Daily Papers

Deep learning approaches have provided state-of-the-art performance in many applications by relying on large and overparameterized neural networks. However, such networks have been shown to be very brittle and are difficult to deploy on resource-limited platforms. Model pruning, i.e., reducing the size of the network, is a widely adopted strategy that can lead to a more robust and compact model. Many heuristics exist for model pruning, but empirical studies show that some heuristics improve performance whereas others can make models more brittle or have other side effects. This work aims to shed light on how different pruning methods alter the network's internal feature representation and the corresponding impact on model performance. To facilitate a comprehensive comparison and characterization of the high-dimensional model feature space, we introduce a visual geometric analysis of feature representations. We decomposed and evaluated a set of critical geometric concepts from the common adopted classification loss, and used them to design a visualization system to compare and highlight the impact of pruning on model performance and feature representation. The proposed tool provides an environment for in-depth comparison of pruning methods and a comprehensive understanding of how model response to common data corruption. By leveraging the proposed visualization, machine learning researchers can reveal the similarities between pruning methods and redundant in robustness evaluation benchmarks, obtain geometric insights about the differences between pruned models that achieve superior robustness performance, and identify samples that are robust or fragile to model pruning and common data corruption to model pruning and data corruption but also obtain insights and explanations on how some pruned models achieve superior robustness performance.

In this work, we study an integrated fault detection and classification framework called FARM for fast, accurate, and robust online chemical process monitoring. The FARM framework integrates the latest advancements in statistical process control (SPC) for monitoring nonparametric and heterogeneous data streams with novel data analysis approaches based on Riemannian geometry together in a hierarchical framework for online process monitoring. We conduct a systematic evaluation of the FARM monitoring framework using the Tennessee Eastman Process (TEP) dataset. Results show that FARM performs competitively against state-of-the-art process monitoring algorithms by achieving a good balance among fault detection rate (FDR), fault detection speed (FDS), and false alarm rate (FAR). Specifically, FARM achieved an average FDR of 96.97% while also outperforming benchmark methods in successfully detecting hard-to-detect faults that are previously known, including Faults 3, 9 and 15, with FDRs being 97.08%, 96.30% and 95.99%, respectively. In terms of FAR, our FARM framework allows practitioners to customize their choice of FAR, thereby offering great flexibility. Moreover, we report a significant improvement in average fault classification accuracy during online monitoring from 61% to 82% when leveraging Riemannian geometric analysis, and further to 84.5% when incorporating additional features from SPC. This illustrates the synergistic effect of integrating fault detection and classification in a holistic, hierarchical monitoring framework.

Multi-Agent Reinforcement Learning (MARL) is increasingly deployed in safety-critical domains, yet methods for interpretable failure detection and attribution remain underdeveloped. We introduce a two-stage gradient-based framework that provides interpretable diagnostics for three critical failure analysis tasks: (1) detecting the true initial failure source (Patient-0); (2) validating why non-attacked agents may be flagged first due to domino effects; and (3) tracing how failures propagate through learned coordination pathways. Stage 1 performs interpretable per-agent failure detection via Taylor-remainder analysis of policy-gradient costs, declaring an initial Patient-0 candidate at the first threshold crossing. Stage 2 provides validation through geometric analysis of critic derivatives-first-order sensitivity and directional second-order curvature aggregated over causal windows to construct interpretable contagion graphs. This approach explains "downstream-first" detection anomalies by revealing pathways that amplify upstream deviations. Evaluated across 500 episodes in Simple Spread (3 and 5 agents) and 100 episodes in StarCraft II using MADDPG and HATRPO, our method achieves 88.2-99.4% Patient-0 detection accuracy while providing interpretable geometric evidence for detection decisions. By moving beyond black-box detection to interpretable gradient-level forensics, this framework offers practical tools for diagnosing cascading failures in safety-critical MARL systems.

The quantum geometric tensor (QGT) is a fundamental quantity for characterizing the geometric properties of quantum states and plays an essential role in elucidating various physical phenomena. The traditional QGT, defined only for pure states, has limited applicability in realistic scenarios where mixed states are common. To address this limitation, we generalize the definition of the QGT to mixed states using the purification bundle and the covariant derivative. Notably, our proposed definition reduces to the traditional QGT when mixed states approach pure states. In our framework, the real and imaginary parts of this generalized QGT correspond to the Bures metric and the mean gauge curvature, respectively, endowing it with a broad range of potential applications. Additionally, using our proposed mixed-state QGT (MSQGT), we derive the geodesic equation applicable to mixed states. This work establishes a unified framework for the geometric analysis of both pure and mixed states, thereby deepening our understanding of the geometric properties of quantum states.

Continuous image editing aims to provide slider-style control of edit strength while preserving source-image fidelity and maintaining a consistent edit direction. Existing learning-based slider methods typically rely on auxiliary modules trained with synthetic or proxy supervision. This introduces additional training overhead and couples slider behavior to the training distribution, which can reduce reliability under distribution shifts in edits or domains. We propose FlowSlider, a training-free method for continuous editing in Rectified Flow that requires no post-training. FlowSlider decomposes FlowEdit's update into (i) a fidelity term, which acts as a source-conditioned stabilizer that preserves identity and structure, and (ii) a steering term that drives semantic transition toward the target edit. Geometric analysis and empirical measurements show that these terms are approximately orthogonal, enabling stable strength control by scaling only the steering term while keeping the fidelity term unchanged. As a result, FlowSlider provides smooth and reliable control without post-training, improving continuous editing quality across diverse tasks.

Adversarial examples are often cited by neuroscientists and machine learning researchers as an example of how computational models diverge from biological sensory systems. Recent work has proposed adding biologically-inspired components to visual neural networks as a way to improve their adversarial robustness. One surprisingly effective component for reducing adversarial vulnerability is response stochasticity, like that exhibited by biological neurons. Here, using recently developed geometrical techniques from computational neuroscience, we investigate how adversarial perturbations influence the internal representations of standard, adversarially trained, and biologically-inspired stochastic networks. We find distinct geometric signatures for each type of network, revealing different mechanisms for achieving robust representations. Next, we generalize these results to the auditory domain, showing that neural stochasticity also makes auditory models more robust to adversarial perturbations. Geometric analysis of the stochastic networks reveals overlap between representations of clean and adversarially perturbed stimuli, and quantitatively demonstrates that competing geometric effects of stochasticity mediate a tradeoff between adversarial and clean performance. Our results shed light on the strategies of robust perception utilized by adversarially trained and stochastic networks, and help explain how stochasticity may be beneficial to machine and biological computation.

Modern large language models (LLMs) such as GPT, Claude, and Gemini have transformed the way we learn, work, and communicate. Yet, their ability to produce highly human-like text raises serious concerns about misinformation and academic integrity, making it an urgent need for reliable algorithms to detect LLM-generated content. In this paper, we start by presenting a geometric approach to demystify rewrite-based detection algorithms, revealing their underlying rationale and demonstrating their generalization ability. Building on this insight, we introduce a novel rewrite-based detection algorithm that adaptively learns the distance between the original and rewritten text. Theoretically, we demonstrate that employing an adaptively learned distance function is more effective for detection than using a fixed distance. Empirically, we conduct extensive experiments with over 100 settings, and find that our approach demonstrates superior performance over baseline algorithms in the majority of scenarios. In particular, it achieves relative improvements from 57.8\% to 80.6\% over the strongest baseline across different target LLMs (e.g., GPT, Claude, and Gemini).

Understanding how Large Language Models (LLMs) perform complex reasoning and their failure mechanisms is a challenge in interpretability research. To provide a measurable geometric analysis perspective, we define the concept of the Reasoning Manifold, a latent low-dimensional geometric structure formed by the internal representations corresponding to all correctly reasoned generations. This structure can be conceptualized as the embodiment of the effective thinking paths that the model has learned to successfully solve a given task. Based on this concept, we build REMA, a framework that explains the origins of failures by quantitatively comparing the spatial relationships of internal model representations corresponding to both erroneous and correct reasoning samples. Specifically, REMA first quantifies the geometric deviation of each erroneous representation by calculating its k-nearest neighbors distance to the approximated manifold formed by correct representations, thereby providing a unified failure signal. It then localizes the divergence points where these deviations first become significant by tracking this deviation metric across the model's layers and comparing it against a baseline of internal fluctuations from correct representations, thus identifying where the reasoning chain begins to go off-track. Our extensive experiments on diverse language and multimodal models and tasks demonstrate the low-dimensional nature of the reasoning manifold and the high separability between erroneous and correct reasoning representations. The results also validate the effectiveness of the REMA framework in analyzing the origins of reasoning failures. This research connects abstract reasoning failures to measurable geometric deviations in representations, providing new avenues for in-depth understanding and diagnosis of the internal computational processes of black-box models.

Enabling agents to understand and interact with complex 3D scenes is a fundamental challenge for embodied artificial intelligence systems. While Multimodal Large Language Models (MLLMs) have achieved significant progress in 2D image understanding, extending such capabilities to 3D scenes remains difficult: 1) 3D environment involves richer concepts such as spatial relationships, affordances, physics, layout, and so on, 2) the absence of large-scale 3D vision-language datasets has posed a significant obstacle. In this paper, we introduce Text-Scene, a framework that automatically parses 3D scenes into textual descriptions for scene understanding. Given a 3D scene, our model identifies object attributes and spatial relationships, and then generates a coherent summary of the whole scene, bridging the gap between 3D observation and language without requiring human-in-the-loop intervention. By leveraging both geometric analysis and MLLMs, Text-Scene produces descriptions that are accurate, detailed, and human-interpretable, capturing object-level details and global-level context. Experimental results on benchmarks demonstrate that our textual parses can faithfully represent 3D scenes and benefit downstream tasks. To evaluate the reasoning capability of MLLMs, we present InPlan3D, a comprehensive benchmark for 3D task planning, consisting of 3174 long-term planning tasks across 636 indoor scenes. We emphasize clarity and accessibility in our approach, aiming to make 3D scene content understandable through language. Code and datasets will be released.

Adversarial prompts capable of jailbreaking large language models (LLMs) and inducing undesirable behaviours pose a significant obstacle to their safe deployment. Current mitigation strategies rely on activating built-in defence mechanisms or fine-tuning the LLMs, but the fundamental distinctions between adversarial and benign prompts are yet to be understood. In this work, we introduce CurvaLID, a novel defense framework that efficiently detects adversarial prompts by leveraging their geometric properties. It is agnostic to the type of LLM, offering a unified detection framework across diverse adversarial prompts and LLM architectures. CurvaLID builds on the geometric analysis of text prompts to uncover their underlying differences. We theoretically extend the concept of curvature via the Whewell equation into an n-dimensional word embedding space, enabling us to quantify local geometric properties, including semantic shifts and curvature in the underlying manifolds. Additionally, we employ Local Intrinsic Dimensionality (LID) to capture geometric features of text prompts within adversarial subspaces. Our findings reveal that adversarial prompts differ fundamentally from benign prompts in terms of their geometric characteristics. Our results demonstrate that CurvaLID delivers superior detection and rejection of adversarial queries, paving the way for safer LLM deployment. The source code can be found at https://github.com/Cancanxxx/CurvaLID

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Data-driven methods such as reinforcement and imitation learning have achieved remarkable success in robot autonomy. However, their data-centric nature still hinders them from generalizing well to ever-changing environments. Moreover, collecting large datasets for robotic tasks is often impractical and expensive. To overcome these challenges, we introduce a new self-supervised neural-symbolic (NeSy) computational framework, imperative learning (IL), for robot autonomy, leveraging the generalization abilities of symbolic reasoning. The framework of IL consists of three primary components: a neural module, a reasoning engine, and a memory system. We formulate IL as a special bilevel optimization (BLO), which enables reciprocal learning over the three modules. This overcomes the label-intensive obstacles associated with data-driven approaches and takes advantage of symbolic reasoning concerning logical reasoning, physical principles, geometric analysis, etc. We discuss several optimization techniques for IL and verify their effectiveness in five distinct robot autonomy tasks including path planning, rule induction, optimal control, visual odometry, and multi-robot routing. Through various experiments, we show that IL can significantly enhance robot autonomy capabilities and we anticipate that it will catalyze further research across diverse domains.

We present ShapeNet: a richly-annotated, large-scale repository of shapes represented by 3D CAD models of objects. ShapeNet contains 3D models from a multitude of semantic categories and organizes them under the WordNet taxonomy. It is a collection of datasets providing many semantic annotations for each 3D model such as consistent rigid alignments, parts and bilateral symmetry planes, physical sizes, keywords, as well as other planned annotations. Annotations are made available through a public web-based interface to enable data visualization of object attributes, promote data-driven geometric analysis, and provide a large-scale quantitative benchmark for research in computer graphics and vision. At the time of this technical report, ShapeNet has indexed more than 3,000,000 models, 220,000 models out of which are classified into 3,135 categories (WordNet synsets). In this report we describe the ShapeNet effort as a whole, provide details for all currently available datasets, and summarize future plans.

We introduce Orion, a visual agent framework that can take in any modality and generate any modality. Using an agentic framework with multiple tool-calling capabilities, Orion is designed for visual AI tasks and achieves state-of-the-art results. Unlike traditional vision-language models that produce descriptive outputs, Orion orchestrates a suite of specialized computer vision tools, including object detection, keypoint localization, panoptic segmentation, Optical Character Recognition, and geometric analysis, to execute complex multi-step visual workflows. The system achieves competitive performance on MMMU, MMBench, DocVQA, and MMLongBench while extending monolithic vision-language models to production-grade visual intelligence. By combining neural perception with symbolic execution, Orion enables autonomous visual reasoning, marking a transition from passive visual understanding to active, tool-driven visual intelligence.

The recent integration of deep learning and pairwise similarity annotation-based constrained clustering -- i.e., deep constrained clustering (DCC) -- has proven effective for incorporating weak supervision into massive data clustering: Less than 1% of pair similarity annotations can often substantially enhance the clustering accuracy. However, beyond empirical successes, there is a lack of understanding of DCC. In addition, many DCC paradigms are sensitive to annotation noise, but performance-guaranteed noisy DCC methods have been largely elusive. This work first takes a deep look into a recently emerged logistic loss function of DCC, and characterizes its theoretical properties. Our result shows that the logistic DCC loss ensures the identifiability of data membership under reasonable conditions, which may shed light on its effectiveness in practice. Building upon this understanding, a new loss function based on geometric factor analysis is proposed to fend against noisy annotations. It is shown that even under unknown annotation confusions, the data membership can still be provably identified under our proposed learning criterion. The proposed approach is tested over multiple datasets to validate our claims.

Reliable depth estimation under real optical conditions remains a core challenge for camera vision in systems such as autonomous robotics and augmented reality. Despite recent progress in depth estimation and depth-of-field rendering, research remains constrained by the lack of large-scale, high-fidelity, real stereo DSLR datasets, limiting real-world generalization and evaluation of models trained on synthetic data as shown extensively in literature. We present the first high-resolution (5472times3648px) stereo DSLR dataset with 18000 images, systematically varying focal length and aperture across complex real scenes and capturing the optical realism and complexity of professional camera systems. For 9 scenes with varying scene complexity, lighting and background, images are captured with two identical camera assemblies at 10 focal lengths (28-70mm) and 5 apertures (f/2.8-f/22), spanning 50 optical configurations in 2000 images per scene. This full-range optics coverage enables controlled analysis of geometric and optical effects for monocular and stereo depth estimation, shallow depth-of-field rendering, deblurring, 3D scene reconstruction and novel view synthesis. Each focal configuration has a dedicated calibration image set, supporting evaluation of classical and learning based methods for intrinsic and extrinsic calibration. The dataset features challenging visual elements such as multi-scale optical illusions, reflective surfaces, mirrors, transparent glass walls, fine-grained details, and natural / artificial ambient light variations. This work attempts to bridge the realism gap between synthetic training data and real camera optics, and demonstrates challenges with the current state-of-the-art monocular, stereo depth and depth-of-field methods. We release the dataset, calibration files, and evaluation code to support reproducible research on real-world optical generalization.

Despite their empirical success, pushing Transformer architectures to extreme depth often leads to a paradoxical failure: representations become increasingly redundant, lose rank, and ultimately collapse. Existing explanations largely attribute this phenomenon to optimization instability or vanishing gradients, yet such accounts fail to explain why collapse persists even under modern normalization and initialization schemes. In this paper, we argue that the collapse of deep Transformers is fundamentally a geometric problem. Standard residual updates implicitly assume that feature accumulation is always beneficial, but offer no mechanism to constrain update directions or to erase outdated information. As depth increases, this leads to systematic drift off the semantic manifold and monotonic feature accumulation, causing representational degeneracy. We propose a unified geometric framework that addresses these failures through two orthogonal principles. First, manifold-constrained hyper-connections restrict residual updates to valid local tangent directions, preventing uncontrolled manifold drift. Second, deep delta learning introduces data-dependent, non-monotonic updates that enable reflection and erasure of redundant features rather than their unconditional accumulation. Together, these mechanisms decouple the direction and sign of feature updates, yielding a stable geometric evolution across depth. We term the resulting architecture the Manifold-Geometric Transformer (MGT). Our analysis predicts that enforcing geometric validity while allowing dynamic erasure is essential for avoiding rank collapse in ultra-deep networks. We outline an evaluation protocol for Transformers exceeding 100 layers to test the hypothesis that geometry, rather than depth itself, is the key limiting factor in deep representation learning.

Analysis of learned representations has a blind spot: it focuses on similarity, measuring how closely embeddings align with external references, but similarity reveals only what is represented, not whether that structure is robust. We introduce geometric stability, a distinct dimension that quantifies how reliably representational geometry holds under perturbation, and present Shesha, a framework for measuring it. Across 2,463 configurations in seven domains, we show that stability and similarity are empirically uncorrelated (ρapprox 0.01) and mechanistically distinct: similarity metrics collapse after removing the top principal components, while stability retains sensitivity to fine-grained manifold structure. This distinction yields actionable insights: for safety monitoring, stability acts as a functional geometric canary, detecting structural drift nearly 2times more sensitively than CKA while filtering out the non-functional noise that triggers false alarms in rigid distance metrics; for controllability, supervised stability predicts linear steerability (ρ= 0.89-0.96); for model selection, stability dissociates from transferability, revealing a geometric tax that transfer optimization incurs. Beyond machine learning, stability predicts CRISPR perturbation coherence and neural-behavioral coupling. By quantifying how reliably systems maintain structure, geometric stability provides a necessary complement to similarity for auditing representations across biological and computational systems.

Representational analysis explores how input data of a neural system are encoded in high dimensional spaces of its distributed neural activations, and how we can compare different systems, for instance, artificial neural networks and brains, on those grounds. While existing methods offer important insights, they typically do not account for local intrinsic geometrical properties within the high-dimensional representation spaces. To go beyond these limitations, we explore Ollivier-Ricci curvature and Ricci flow as tools to study the alignment of representations between humans and artificial neural systems on a geometric level. As a proof-of-principle study, we compared the representations of face stimuli between VGG-Face, a human-aligned version of VGG-Face, and corresponding human similarity judgments from a large online study. Using this discrete geometric framework, we were able to identify local structural similarities and differences by examining the distributions of node and edge curvature and higher-level properties by detecting and comparing community structure in the representational graphs.

Retargeting human motion to robot poses is a practical approach for teleoperating bimanual humanoid robot arms, but existing methods can be suboptimal and slow, often causing undesirable motion or latency. This is due to optimizing to match robot end-effector to human hand position and orientation, which can also limit the robot's workspace to that of the human. Instead, this paper reframes retargeting as an orientation alignment problem, enabling a closed-form, geometric solution algorithm with an optimality guarantee. The key idea is to align a robot arm to a human's upper and lower arm orientations, as identified from shoulder, elbow, and wrist (SEW) keypoints; hence, the method is called SEW-Mimic. The method has fast inference (3 kHz) on standard commercial CPUs, leaving computational overhead for downstream applications; an example in this paper is a safety filter to avoid bimanual self-collision. The method suits most 7-degree-of-freedom robot arms and humanoids, and is agnostic to input keypoint source. Experiments show that SEW-Mimic outperforms other retargeting methods in computation time and accuracy. A pilot user study suggests that the method improves teleoperation task success. Preliminary analysis indicates that data collected with SEW-Mimic improves policy learning due to being smoother. SEW-Mimic is also shown to be a drop-in way to accelerate full-body humanoid retargeting. Finally, hardware demonstrations illustrate SEW-Mimic's practicality. The results emphasize the utility of SEW-Mimic as a fundamental building block for bimanual robot manipulation and humanoid robot teleoperation.

When retrieval-augmented generation (RAG) systems hallucinate, what geometric trace does this leave in embedding space? We introduce the Semantic Grounding Index (SGI), defined as the ratio of angular distances from the response to the question versus the context on the unit hypersphere S^{d-1}.Our central finding is semantic laziness: hallucinated responses remain angularly proximate to questions rather than departing toward retrieved contexts. On HaluEval (n=5,000), we observe large effect sizes (Cohen's d ranging from 0.92 to 1.28) across five embedding models with mean cross-model correlation r=0.85. Crucially, we derive from the spherical triangle inequality that SGI's discriminative power should increase with question-context angular separation θ(q,c)-a theoretical prediction confirmed empirically: effect size rises monotonically from d=0.61 -low θ(q,c), to d=1.27 -high θ(q,c), with AUC improving from 0.72 to 0.83. Subgroup analysis reveals that SGI excels on long responses (d=2.05) and short questions (d=1.22), while remaining robust across context lengths. Calibration analysis yields ECE=0.10, indicating SGI scores can serve as probability estimates, not merely rankings. A critical negative result on TruthfulQA (AUC=0.478) establishes that angular geometry measures topical engagement rather than factual accuracy. SGI provides computationally efficient, theoretically grounded infrastructure for identifying responses that warrant verification in production RAG deployments.

Graph Neural Networks (GNN) can capture the geometric properties of neural representations in EEG data. Here we utilise those to study how reinforcement-based motor learning affects neural activity patterns during motor planning, leveraging the inherent graph structure of EEG channels to capture the spatial relationships in brain activity. By exploiting task-specific symmetries, we define different pretraining strategies that not only improve model performance across all participant groups but also validate the robustness of the geometric representations. Explainability analysis based on the graph structures reveals consistent group-specific neural signatures that persist across pretraining conditions, suggesting stable geometric structures in the neural representations associated with motor learning and feedback processing. These geometric patterns exhibit partial invariance to certain task space transformations, indicating symmetries that enable generalisation across conditions while maintaining specificity to individual learning strategies. This work demonstrates how GNNs can uncover the effects of previous outcomes on motor planning, in a complex real-world task, providing insights into the geometric principles governing neural representations. Our experimental design bridges the gap between controlled experiments and ecologically valid scenarios, offering new insights into the organisation of neural representations during naturalistic motor learning, which may open avenues for exploring fundamental principles governing brain activity in complex tasks.

The advancement of large language models (LLMs) for real-world applications hinges critically on enhancing their reasoning capabilities. In this work, we explore the reasoning abilities of large language models (LLMs) through their geometrical understanding. We establish a connection between the expressive power of LLMs and the density of their self-attention graphs. Our analysis demonstrates that the density of these graphs defines the intrinsic dimension of the inputs to the MLP blocks. We demonstrate through theoretical analysis and toy examples that a higher intrinsic dimension implies a greater expressive capacity of the LLM. We further provide empirical evidence linking this geometric framework to recent advancements in methods aimed at enhancing the reasoning capabilities of LLMs.

We introduce a novel approach for analyzing the training dynamics of ReLU networks by examining the characteristic activation boundaries of individual ReLU neurons. Our proposed analysis reveals a critical instability in common neural network parameterizations and normalizations during stochastic optimization, which impedes fast convergence and hurts generalization performance. Addressing this, we propose Geometric Parameterization (GmP), a novel neural network parameterization technique that effectively separates the radial and angular components of weights in the hyperspherical coordinate system. We show theoretically that GmP resolves the aforementioned instability issue. We report empirical results on various models and benchmarks to verify GmP's theoretical advantages of optimization stability, convergence speed and generalization performance.

Evaluating the symbolic reasoning of large language models (LLMs) calls for geometry benchmarks that require multi-step proofs grounded in both text and diagrams. However, existing benchmarks are often limited in scale and rarely provide visually grounded multiple-choice questions, limiting reliable evaluation of complex reasoning. We introduce GeoChallenge, a dataset of 90K automatically generated multiple-choice geometry proof problems, each requiring multi-step reasoning over aligned textual descriptions and diagrams. GeoChallenge provides fine-grained complexity ratings and formal language annotations to enable controlled evaluation. Experiments on multiple advanced LLMs show a clear performance gap between models and humans (the best-performing model, GPT-5-nano, achieves 75.89 exact match vs. 94.74 for humans). Further analysis also reveals three common failure patterns of LLMs: (1) exact match failures under the multiple-choice setting; (2) weak visual reliance; and (3) overextended reasoning without convergence.

Recent advancements, such as Group Relative Policy Optimization (GRPO), have enhanced the reasoning capabilities of large language models by optimizing the arithmetic mean of token-level rewards. However, GRPO suffers from unstable policy updates when processing tokens with outlier importance-weighted rewards, which manifests as extreme importance sampling ratios during training, i.e., the ratio between the sampling probabilities assigned to a token by the current and old policies. In this work, we propose Geometric-Mean Policy Optimization (GMPO), a stabilized variant of GRPO. Instead of optimizing the arithmetic mean, GMPO maximizes the geometric mean of token-level rewards, which is inherently less sensitive to outliers and maintains a more stable range of importance sampling ratio. In addition, we provide comprehensive theoretical and experimental analysis to justify the design and stability benefits of GMPO. Beyond improved stability, GMPO-7B outperforms GRPO by an average of 4.1% on multiple mathematical benchmarks and 1.4% on multimodal reasoning benchmark, including AIME24, AMC, MATH500, OlympiadBench, Minerva, and Geometry3K. Code is available at https://github.com/callsys/GMPO.

Errors in understanding visual information in images (i.e., visual perception errors) remain a major source of mistakes in Large Vision Language Models (LVLMs). While further analysis is essential, there is a deficiency in datasets for evaluating the visual perception of LVLMs. In this work, we introduce VisOnlyQA, a new dataset designed to directly evaluate the visual perception capabilities of LVLMs on questions about geometric and numerical information in scientific figures. Our dataset enables us to analyze the visual perception of LVLMs for fine-grained visual information, independent of other capabilities such as reasoning. The evaluation set of VisOnlyQA includes 1,200 multiple-choice questions in 12 tasks on four categories of figures. We also provide synthetic training data consisting of 70k instances. Our experiments on VisOnlyQA highlight the following findings: (i) 20 LVLMs we evaluate, including GPT-4o and Gemini 1.5 Pro, work poorly on the visual perception tasks in VisOnlyQA, while human performance is nearly perfect. (ii) Fine-tuning on synthetic training data demonstrates the potential for enhancing the visual perception of LVLMs, but observed improvements are limited to certain tasks and specific models. (iii) Stronger language models improve the visual perception of LVLMs. In summary, our experiments suggest that both training data and model architectures should be improved to enhance the visual perception capabilities of LVLMs. The datasets, code, and model responses are provided at https://github.com/psunlpgroup/VisOnlyQA.

Geometry problem-solving (GPS), a challenging task requiring both visual comprehension and symbolic reasoning, effectively measures the reasoning capabilities of multimodal large language models (MLLMs). Humans exhibit strong reasoning ability in this task through accurate identification and adaptive application of geometric principles within visual contexts. However, existing benchmarks fail to jointly assess both dimensions of the human-like geometric reasoning mechanism in MLLMs, remaining a critical gap in assessing their ability to tackle GPS. To this end, we introduce GeoSense, the first comprehensive bilingual benchmark designed to systematically evaluate the geometric reasoning abilities of MLLMs through the lens of geometric principles. GeoSense features a five-level hierarchical framework of geometric principles spanning plane and solid geometry, an intricately annotated dataset of 1,789 problems, and an innovative evaluation strategy. Through extensive experiments on GeoSense with various open-source and closed-source MLLMs, we observe that Gemini-2.0-pro-flash performs best, achieving an overall score of 65.3. Our in-depth analysis reveals that the identification and application of geometric principles remain a bottleneck for leading MLLMs, jointly hindering their reasoning abilities. These findings underscore GeoSense's potential to guide future advancements in MLLMs' geometric reasoning capabilities, paving the way for more robust and human-like reasoning in artificial intelligence.

For a language model (LM) to faithfully model human language, it must compress vast, potentially infinite information into relatively few dimensions. We propose analyzing compression in (pre-trained) LMs from two points of view: geometric and information-theoretic. We demonstrate that the two views are highly correlated, such that the intrinsic geometric dimension of linguistic data predicts their coding length under the LM. We then show that, in turn, high compression of a linguistic dataset predicts rapid adaptation to that dataset, confirming that being able to compress linguistic information is an important part of successful LM performance. As a practical byproduct of our analysis, we evaluate a battery of intrinsic dimension estimators for the first time on linguistic data, showing that only some encapsulate the relationship between information-theoretic compression, geometric compression, and ease-of-adaptation.

Geometric deep learning, i.e., designing neural networks to handle the ubiquitous geometric data such as point clouds and graphs, have achieved great successes in the last decade. One critical inductive bias is that the model can maintain invariance towards various transformations such as translation, rotation, and scaling. The existing graph neural network (GNN) approaches can only maintain permutation-invariance, failing to guarantee invariance with respect to other transformations. Besides GNNs, other works design sophisticated transformation-invariant layers, which are computationally expensive and difficult to be extended. To solve this problem, we revisit why the existing neural networks cannot maintain transformation invariance when handling geometric data. Our findings show that transformation-invariant and distance-preserving initial representations are sufficient to achieve transformation invariance rather than needing sophisticated neural layer designs. Motivated by these findings, we propose Transformation Invariant Neural Networks (TinvNN), a straightforward and general framework for geometric data. Specifically, we realize transformation-invariant and distance-preserving initial point representations by modifying multi-dimensional scaling before feeding the representations into neural networks. We prove that TinvNN can strictly guarantee transformation invariance, being general and flexible enough to be combined with the existing neural networks. Extensive experimental results on point cloud analysis and combinatorial optimization demonstrate the effectiveness and general applicability of our proposed method. Based on the experimental results, we advocate that TinvNN should be considered a new starting point and an essential baseline for further studies of transformation-invariant geometric deep learning.

This paper investigates the performance of network non-orthogonal multiple access (N-NOMA) in a downlink coordinated multi-point (CoMP) system. In the considered N-NOMA scheme, multiple base stations (BSs) cooperatively serve a CoMP user, meanwhile, each BS serves additional NOMA users by occupying the same resource block allocated to the CoMP user. The locations of the BSs and users are modeled by stochastic geometric models and the interference from the whole network is considered. Through rigorous derivations, the outage probabilities achieved by the CoMP and NOMA users are obtained, respectively. Numerical results are provided to verify the accuracy of the analytical results and also demonstrate the superior performance of N-NOMA compared to orthogonal multiple access (OMA) based CoMP scheme.

We construct a geometric framework for cosmological large-scale structure based on optimal transport theory and Wasserstein geometry. In this framework, Ricci curvature on the probability measure space P_2(M) is characterized by the geodesic convexity of entropy and is formulated as the response of probability distributions to optimal transport. We introduce effective Ricci curvatures K_{eff}^{(infty)} and K_{eff}^{(N)} associated with Kullback--Leibler-type and Rényi-type entropies, corresponding respectively to the curvature-dimension conditions CD(K,infty) and CD(K,N). By localizing these curvatures to finite scales using local and reference measures, we construct curvature indicators applicable to observational data. Under a local quadratic approximation, the effective curvature reduces to the Hessian of the log-density, showing that conventional Hessian-based structure classifications arise as a limiting case of the present framework. We further show that effective curvature depends on observational scale and formulate this dependence as a scale flow, distinct from Ricci flow because it describes a change of resolution rather than a time evolution of geometry. Treating curvature as a random field then extends the statistical description of density fields: curvature statistics are given by higher-order weighted integrals of the power spectrum and by spatial derivatives of the correlation function, emphasizing geometric rather than amplitude information. This framework provides a unified connection between optimal transport geometry and cosmological structure analysis, and offers a new perspective on multiscale structure and nonlinear statistics.

Machine unlearning, the removal of a training subset's influence from a deployed model, is critical for privacy preservation and model reliability, yet gradient ascent on forget samples often harms retained knowledge. Existing approaches face a persistent tradeoff between effective forgetting and preservation on the retain set. While previous methods provide useful heuristics, they often lack a formal analysis on how exactly forgetting updates harm retained knowledge, and whether the side effects can be removed with theoretical guarantees. To explore a theoretically sound and simple solution, we start from the first principle on how performance on the retain set is actually affected: a first-order analysis of the local change of the retain loss under small parameter updates during model training. We start from a crisp equivalence: the retain loss is unchanged to first order iff the update direction is orthogonal to the subspace spanned by retain gradients ("retain-invariant"). This identifies the entangled component as the tangential part of forget update within the retain-gradient subspace, and characterizes disentanglement as orthogonality. Guided by this, we propose the Geometric-disentanglement Unlearning (GU) that decomposes any candidate forget gradient update into tangential and normal components to retain space and executes only the normal component. Under a standard trust-region budget, the projected direction aligned with the raw forget gradient is optimal among all first-order retain-invariant moves, and we also derive the optimal projected direction for joint forget-retain updating objectives. Our method is plug-and-play and can be attached to existing gradient-based unlearning procedures to mitigate side effects. GU achieves consistent improvement on various methods across three benchmarks TOFU, MUSE, and WMDP.

This paper bridges internal and external analysis approaches to large language models (LLMs) by demonstrating that geometric properties of internal model representations serve as reliable proxies for evaluating generated text quality. We validate a set of metrics including Maximum Explainable Variance, Effective Rank, Intrinsic Dimensionality, MAUVE score, and Schatten Norms measured across different layers of LLMs, demonstrating that Intrinsic Dimensionality and Effective Rank can serve as universal assessments of text naturalness and quality. Our key finding reveals that different models consistently rank text from various sources in the same order based on these geometric properties, indicating that these metrics reflect inherent text characteristics rather than model-specific artifacts. This allows a reference-free text quality evaluation that does not require human-annotated datasets, offering practical advantages for automated evaluation pipelines.

We introduce randomized algorithms to Clifford's Geometric Algebra, generalizing randomized linear algebra to hypercomplex vector spaces. This novel approach has many implications in machine learning, including training neural networks to global optimality via convex optimization. Additionally, we consider fine-tuning large language model (LLM) embeddings as a key application area, exploring the intersection of geometric algebra and modern AI techniques. In particular, we conduct a comparative analysis of the robustness of transfer learning via embeddings, such as OpenAI GPT models and BERT, using traditional methods versus our novel approach based on convex optimization. We test our convex optimization transfer learning method across a variety of case studies, employing different embeddings (GPT-4 and BERT embeddings) and different text classification datasets (IMDb, Amazon Polarity Dataset, and GLUE) with a range of hyperparameter settings. Our results demonstrate that convex optimization and geometric algebra not only enhances the performance of LLMs but also offers a more stable and reliable method of transfer learning via embeddings.

The inference of topological principles is a key problem in structured reconstruction. We observe that wrongly predicted topological relationships are often incurred by the lack of holistic geometry clues in low-level features. Inspired by the fact that massive signals can be compactly described with frequency analysis, we experimentally explore the efficiency and tendency of learning structure geometry in the frequency domain. Accordingly, we propose a frequency-domain feature learning strategy (F-Learn) to fuse scattered geometric fragments holistically for topology-intact structure reasoning. Benefiting from the parsimonious design, the F-Learn strategy can be easily deployed into a deep reconstructor with a lightweight model modification. Experiments demonstrate that the F-Learn strategy can effectively introduce structure awareness into geometric primitive detection and topology inference, bringing significant performance improvement to final structured reconstruction. Code and pre-trained models are available at https://github.com/Geo-Tell/F-Learn.

Introduction: Mechanical thrombectomy can cause vessel deformation and procedure-related injury. Benchtop models are widely used for device testing, but time-resolved, full-field 3D vessel-motion measurements remain limited. Methods: We developed a nine-camera, low-cost multi-view workflow for benchtop thrombectomy in silicone middle cerebral artery phantoms (2160p, 20 fps). Multi-view videos were calibrated, segmented, and reconstructed with 4D Gaussian Splatting. Reconstructed point clouds were converted to fixed-connectivity edge graphs for region-of-interest (ROI) displacement tracking and a relative surface-based stress proxy. Stress-proxy values were derived from edge stretch using a Neo-Hookean mapping and reported as comparative surface metrics. A synthetic Blender pipeline with known deformation provided geometric and temporal validation. Results: In synthetic bulk translation, the stress proxy remained near zero for most edges (median approx 0 MPa; 90th percentile 0.028 MPa), with sparse outliers. In synthetic pulling (1-5 mm), reconstruction showed close geometric and temporal agreement with ground truth, with symmetric Chamfer distance of 1.714-1.815 mm and precision of 0.964-0.972 at τ= 1 mm. In preliminary benchtop comparative trials (one trial per condition), cervical aspiration catheter placement showed higher max-median ROI displacement and stress-proxy values than internal carotid artery terminus placement. Conclusion: The proposed protocol provides standardized, time-resolved surface kinematics and comparative relative displacement and stress proxy measurements for thrombectomy benchtop studies. The framework supports condition-to-condition comparisons and methods validation, while remaining distinct from absolute wall-stress estimation. Implementation code and example data are available at https://ethanuser.github.io/vessel4D

Differential privacy (DP)'s effect in medical imaging is typically evaluated only through end-to-end performance, leaving the mechanism of privacy-induced utility loss unclear. We introduce Differential Privacy Representation Geometry for Medical Imaging (DP-RGMI), a framework that interprets DP as a structured transformation of representation space and decomposes performance degradation into encoder geometry and task-head utilization. Geometry is quantified by representation displacement from initialization and spectral effective dimension, while utilization is measured as the gap between linear-probe and end-to-end utility. Across over 594,000 images from four chest X-ray datasets and multiple pretrained initializations, we show that DP is consistently associated with a utilization gap even when linear separability is largely preserved. At the same time, displacement and spectral dimension exhibit non-monotonic, initialization- and dataset-dependent reshaping, indicating that DP alters representation anisotropy rather than uniformly collapsing features. Correlation analysis reveals that the association between end-to-end performance and utilization is robust across datasets but can vary by initialization, while geometric quantities capture additional prior- and dataset-conditioned variation. These findings position DP-RGMI as a reproducible framework for diagnosing privacy-induced failure modes and informing privacy model selection.

High-performance real-time stereo matching methods invariably rely on 3D regularization of the cost volume, which is unfriendly to mobile devices. And 2D regularization based methods struggle in ill-posed regions. In this paper, we present a deployment-friendly 4D cost aggregation network DBStereo, which is based on pure 2D convolutions. Specifically, we first provide a thorough analysis of the decoupling characteristics of 4D cost volume. And design a lightweight bidirectional geometry aggregation block to capture spatial and disparity representation respectively. Through decoupled learning, our approach achieves real-time performance and impressive accuracy simultaneously. Extensive experiments demonstrate that our proposed DBStereo outperforms all existing aggregation-based methods in both inference time and accuracy, even surpassing the iterative-based method IGEV-Stereo. Our study break the empirical design of using 3D convolutions for 4D cost volume and provides a simple yet strong baseline of the proposed decouple aggregation paradigm for further study. Code will be available at (https://github.com/happydummy/DBStereo{https://github.com/happydummy/DBStereo}) soon.

We present the wakefield conformal mapping technique that can be readily applied to the analysis of the radiation generated by an ultra-relativistic particle in the step transition and a collimator. We derive simple analytical expressions for the lower and upper bounds of both longitudinal and transverse wake potentials. We test the derived expressions against well-known formulas in several representative examples. The proposed method can greatly simplify the optimization of collimating sections, as well as become a useful tool in the shape optimization problems.

Geometric Attention (GA) specifies an attention layer by four independent inputs: a finite carrier (what indices are addressable), an evidence-kernel rule (how masked proto-scores and a link induce nonnegative weights), a probe family (which observables are treated as admissible), and an anchor/update rule (which representative kernel is selected and how it is applied). Probe families induce an operational equivalence relation on kernels and therefore a gauge; anchors select representatives relative to that probe. Under a scalar relational-work representation and a multiplicative compositionality law for evidence, the admissible link family is exponential, yielding Gibbs weights; with row anchoring this includes the softmax kernel family as a subregime. After quotienting unary row/column score fields, the remaining interaction component admits a canonical rank-r normal form (Eckart-Young/SVD); dot-product score charts implement the corresponding low-rank interaction regime. Fixing the carrier and extensionalizing the update yields the standard fixed-token Transformer attention operator; allowing carrier updates yields adaptive-carrier and staged-depth regimes. The operator language also supports multihead/mixed kernels, plan-based anchors (e.g., entropic OT/Sinkhorn), and unary operators (e.g., FFN-style fields) as explicit regime choices. This separates invariant structure from modeling choice, enabling principled comparison and extension of attention mechanisms, and attention-based architectures.

Spatial reasoning is a core aspect of human intelligence that allows perception, inference and planning in 3D environments. However, current vision-language models (VLMs) struggle to maintain geometric coherence and cross-view consistency for spatial reasoning in multi-view settings. We attribute this gap to the lack of fine-grained benchmarks that isolate multi-view reasoning from single-view perception and temporal factors. To address this, we present ReMindView-Bench, a cognitively grounded benchmark for evaluating how VLMs construct, align and maintain spatial mental models across complementary viewpoints. ReMindView-Bench systematically varies viewpoint spatial pattern and query type to probe key factors of spatial cognition. Evaluations of 15 current VLMs reveals consistent failures in cross-view alignment and perspective-taking in multi-view spatial reasoning, motivating deeper analysis on the reasoning process. Explicit phase-wise analysis using LLM-as-a-judge and self-consistency prompting shows that VLMs perform well on in-frame perception but degrade sharply when integrating information across views. Implicit analysis, including linear probing and entropy dynamics, further show progressive loss of task-relevant information and uncertainty separation between correct and incorrect trajectories. These results provide a cognitively grounded diagnosis of VLM spatial reasoning and reveal how multi-view spatial mental models are formed, degraded and destabilized across reasoning phases. The ReMindView-Bench benchmark is available at https://huggingface.co/datasets/Xue0823/ReMindView-Bench, and the source codes of benchmark construction and VLM reasoning analysis are available at https://github.com/pittisl/ReMindView-Bench.

The scarcity of annotations poses a significant challenge in medical image analysis. Large-scale pre-training has emerged as a promising label-efficient solution, owing to the utilization of large-scale data, large models, and advanced pre-training techniques. However, its development in medical images remains underexplored. The primary challenge lies in harnessing large-scale unlabeled data and learning high-level semantics without annotations. We observe that 3D medical images exhibit consistent geometric context, i.e., consistent geometric relations between different organs, which leads to a promising way for learning consistent representations. Motivated by this, we introduce a simple-yet-effective Volume Contrast (VoCo) framework to leverage geometric context priors for self-supervision. Given an input volume, we extract base crops from different regions to construct positive and negative pairs for contrastive learning. Then we predict the contextual position of a random crop by contrasting its similarity to the base crops. In this way, VoCo encodes the inherent geometric context into model representations, facilitating high-level semantic learning without annotations. Specifically, we (1) introduce the largest medical pre-training dataset PreCT-160K; (2) investigate scaling laws and propose guidelines for tailoring different model sizes to various medical tasks; (3) build a benchmark encompassing 48 medical tasks. Extensive experiments highlight the superiority of VoCo. Codes at https://github.com/Luffy03/Large-Scale-Medical.

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.

Computational RNA design tasks are often posed as inverse problems, where sequences are designed based on adopting a single desired secondary structure without considering 3D geometry and conformational diversity. We introduce gRNAde, a geometric RNA design pipeline operating on 3D RNA backbones to design sequences that explicitly account for structure and dynamics. Under the hood, gRNAde is a multi-state Graph Neural Network that generates candidate RNA sequences conditioned on one or more 3D backbone structures where the identities of the bases are unknown. On a single-state fixed backbone re-design benchmark of 14 RNA structures from the PDB identified by Das et al. [2010], gRNAde obtains higher native sequence recovery rates (56% on average) compared to Rosetta (45% on average), taking under a second to produce designs compared to the reported hours for Rosetta. We further demonstrate the utility of gRNAde on a new benchmark of multi-state design for structurally flexible RNAs, as well as zero-shot ranking of mutational fitness landscapes in a retrospective analysis of a recent ribozyme. Open source code: https://github.com/chaitjo/geometric-rna-design

We present a Non-parametric Network for 3D point cloud analysis, Point-NN, which consists of purely non-learnable components: farthest point sampling (FPS), k-nearest neighbors (k-NN), and pooling operations, with trigonometric functions. Surprisingly, it performs well on various 3D tasks, requiring no parameters or training, and even surpasses existing fully trained models. Starting from this basic non-parametric model, we propose two extensions. First, Point-NN can serve as a base architectural framework to construct Parametric Networks by simply inserting linear layers on top. Given the superior non-parametric foundation, the derived Point-PN exhibits a high performance-efficiency trade-off with only a few learnable parameters. Second, Point-NN can be regarded as a plug-and-play module for the already trained 3D models during inference. Point-NN captures the complementary geometric knowledge and enhances existing methods for different 3D benchmarks without re-training. We hope our work may cast a light on the community for understanding 3D point clouds with non-parametric methods. Code is available at https://github.com/ZrrSkywalker/Point-NN.

Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on a slight simplification of DQN that fully captures its key features. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by DQN. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players. Borrowing the analysis of DQN, we also quantify the difference between the policies obtained by Minimax-DQN and the Nash equilibrium of the Markov game in terms of both the algorithmic and statistical rates of convergence.

Existing Transformer-based models for point cloud analysis suffer from quadratic complexity, leading to compromised point cloud resolution and information loss. In contrast, the newly proposed Mamba model, based on state space models (SSM), outperforms Transformer in multiple areas with only linear complexity. However, the straightforward adoption of Mamba does not achieve satisfactory performance on point cloud tasks. In this work, we present Mamba3D, a state space model tailored for point cloud learning to enhance local feature extraction, achieving superior performance, high efficiency, and scalability potential. Specifically, we propose a simple yet effective Local Norm Pooling (LNP) block to extract local geometric features. Additionally, to obtain better global features, we introduce a bidirectional SSM (bi-SSM) with both a token forward SSM and a novel backward SSM that operates on the feature channel. Extensive experimental results show that Mamba3D surpasses Transformer-based counterparts and concurrent works in multiple tasks, with or without pre-training. Notably, Mamba3D achieves multiple SoTA, including an overall accuracy of 92.6% (train from scratch) on the ScanObjectNN and 95.1% (with single-modal pre-training) on the ModelNet40 classification task, with only linear complexity.

A triangular mesh is one of the most popular 3D data representations. As such, the deployment of deep neural networks for mesh processing is widely spread and is increasingly attracting more attention. However, neural networks are prone to adversarial attacks, where carefully crafted inputs impair the model's functionality. The need to explore these vulnerabilities is a fundamental factor in the future development of 3D-based applications. Recently, mesh attacks were studied on the semantic level, where classifiers are misled to produce wrong predictions. Nevertheless, mesh surfaces possess complex geometric attributes beyond their semantic meaning, and their analysis often includes the need to encode and reconstruct the geometry of the shape. We propose a novel framework for a geometric adversarial attack on a 3D mesh autoencoder. In this setting, an adversarial input mesh deceives the autoencoder by forcing it to reconstruct a different geometric shape at its output. The malicious input is produced by perturbing a clean shape in the spectral domain. Our method leverages the spectral decomposition of the mesh along with additional mesh-related properties to obtain visually credible results that consider the delicacy of surface distortions. Our code is publicly available at https://github.com/StolikTomer/SAGA.

Multimodal Sentiment Analysis (MSA) leverages heterogeneous modalities, such as language, vision, and audio, to enhance the understanding of human sentiment. While existing models often focus on extracting shared information across modalities or directly fusing heterogeneous modalities, such approaches can introduce redundancy and conflicts due to equal treatment of all modalities and the mutual transfer of information between modality pairs. To address these issues, we propose a Disentangled-Language-Focused (DLF) multimodal representation learning framework, which incorporates a feature disentanglement module to separate modality-shared and modality-specific information. To further reduce redundancy and enhance language-targeted features, four geometric measures are introduced to refine the disentanglement process. A Language-Focused Attractor (LFA) is further developed to strengthen language representation by leveraging complementary modality-specific information through a language-guided cross-attention mechanism. The framework also employs hierarchical predictions to improve overall accuracy. Extensive experiments on two popular MSA datasets, CMU-MOSI and CMU-MOSEI, demonstrate the significant performance gains achieved by the proposed DLF framework. Comprehensive ablation studies further validate the effectiveness of the feature disentanglement module, language-focused attractor, and hierarchical predictions. Our code is available at https://github.com/pwang322/DLF.

Cluster analysis relies on effective benchmarks for evaluating and comparing different algorithms. Simulation studies on synthetic data are popular because important features of the data sets, such as the overlap between clusters, or the variation in cluster shapes, can be effectively varied. Unfortunately, creating evaluation scenarios is often laborious, as practitioners must translate higher-level scenario descriptions like "clusters with very different shapes" into lower-level geometric parameters such as cluster centers, covariance matrices, etc. To make benchmarks more convenient and informative, we propose synthetic data generation based on direct specification of high-level scenarios, either through verbal descriptions or high-level geometric parameters. Our open-source Python package repliclust implements this workflow, making it easy to set up interpretable and reproducible benchmarks for cluster analysis. A demo of data generation from verbal inputs is available at https://demo.repliclust.org.

Prior ReLoc3R achieves breakthrough performance with fast 25ms inference and state-of-the-art regression accuracy, yet our analysis reveals subtle geometric inconsistencies in its internal representations that prevent reaching the precision ceiling of correspondence-based methods like MASt3R (which require 300ms per pair). In this work, we present GeLoc3r, a novel approach to relative camera pose estimation that enhances pose regression methods through Geometric Consistency Regularization (GCR). GeLoc3r overcomes the speed-accuracy dilemma by training regression networks to produce geometrically consistent poses without inference-time geometric computation. During training, GeLoc3r leverages ground-truth depth to generate dense 3D-2D correspondences, weights them using a FusionTransformer that learns correspondence importance, and computes geometrically-consistent poses via weighted RANSAC. This creates a consistency loss that transfers geometric knowledge into the regression network. Unlike FAR method which requires both regression and geometric solving at inference, GeLoc3r only uses the enhanced regression head at test time, maintaining ReLoc3R's fast speed and approaching MASt3R's high accuracy. On challenging benchmarks, GeLoc3r consistently outperforms ReLoc3R, achieving significant improvements including 40.45% vs. 34.85% AUC@5° on the CO3Dv2 dataset (16% relative improvement), 68.66% vs. 66.70% AUC@5° on RealEstate10K, and 50.45% vs. 49.60% on MegaDepth1500. By teaching geometric consistency during training rather than enforcing it at inference, GeLoc3r represents a paradigm shift in how neural networks learn camera geometry, achieving both the speed of regression and the geometric understanding of correspondence methods.

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance varepsilon, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as varepsilon varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from varepsilon=0, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of varepsilon. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.

In this paper, we propose a hybrid approach for multi-object microbial cell segmentation. The approach combines an ML-based detection with a geometry-aware variational-based segmentation using B-splines that are parametrized based on a geometric model of the cell shape. The detection is done first using YOLOv5. In a second step, each detected cell is segmented individually. Thus, the segmentation only needs to be done on a per-cell basis, which makes it amenable to a variational approach that incorporates prior knowledge on the geometry. Here, the contour of the segmentation is modelled as closed uniform cubic B-spline, whose control points are parametrized using the known cell geometry. Compared to purely ML-based segmentation approaches, which need accurate segmentation maps as training data that are very laborious to produce, our method just needs bounding boxes as training data. Still, the proposed method performs on par with ML-based segmentation approaches usually used in this context. We study the performance of the proposed method on time-lapse microscopy data of Corynebacterium glutamicum.

Geometric alignment appears in a variety of applications, ranging from domain adaptation, optimal transport, and normalizing flows in machine learning; optical flow and learned augmentation in computer vision and deformable registration within biomedical imaging. A recurring challenge is the alignment of domains whose topology is not the same; a problem that is routinely ignored, potentially introducing bias in downstream analysis. As a first step towards solving such alignment problems, we propose an unsupervised algorithm for the detection of changes in image topology. The model is based on a conditional variational auto-encoder and detects topological changes between two images during the registration step. We account for both topological changes in the image under spatial variation and unexpected transformations. Our approach is validated on two tasks and datasets: detection of topological changes in microscopy images of cells, and unsupervised anomaly detection brain imaging.

Self-supervised learning (SSL) has demonstrated remarkable success in 3D point cloud analysis, particularly through masked autoencoders (MAEs). However, existing MAE-based methods lack rotation invariance, leading to significant performance degradation when processing arbitrarily rotated point clouds in real-world scenarios. To address this limitation, we introduce Handcrafted Feature-Based Rotation-Invariant Masked Autoencoder (HFBRI-MAE), a novel framework that refines the MAE design with rotation-invariant handcrafted features to ensure stable feature learning across different orientations. By leveraging both rotation-invariant local and global features for token embedding and position embedding, HFBRI-MAE effectively eliminates rotational dependencies while preserving rich geometric structures. Additionally, we redefine the reconstruction target to a canonically aligned version of the input, mitigating rotational ambiguities. Extensive experiments on ModelNet40, ScanObjectNN, and ShapeNetPart demonstrate that HFBRI-MAE consistently outperforms existing methods in object classification, segmentation, and few-shot learning, highlighting its robustness and strong generalization ability in real-world 3D applications.

Self-Supervised Learning (SSL) has demonstrated promising results in 3D medical image analysis. However, the lack of high-level semantics in pre-training still heavily hinders the performance of downstream tasks. We observe that 3D medical images contain relatively consistent contextual position information, i.e., consistent geometric relations between different organs, which leads to a potential way for us to learn consistent semantic representations in pre-training. In this paper, we propose a simple-yet-effective Volume Contrast (VoCo) framework to leverage the contextual position priors for pre-training. Specifically, we first generate a group of base crops from different regions while enforcing feature discrepancy among them, where we employ them as class assignments of different regions. Then, we randomly crop sub-volumes and predict them belonging to which class (located at which region) by contrasting their similarity to different base crops, which can be seen as predicting contextual positions of different sub-volumes. Through this pretext task, VoCo implicitly encodes the contextual position priors into model representations without the guidance of annotations, enabling us to effectively improve the performance of downstream tasks that require high-level semantics. Extensive experimental results on six downstream tasks demonstrate the superior effectiveness of VoCo. Code will be available at https://github.com/Luffy03/VoCo.

Empirically validating new 3D-printing related algorithms and implementations requires testing data representative of inputs encountered in the wild. An ideal benchmarking dataset should not only draw from the same distribution of shapes people print in terms of class (e.g., toys, mechanisms, jewelry), representation type (e.g., triangle soup meshes) and complexity (e.g., number of facets), but should also capture problems and artifacts endemic to 3D printing models (e.g., self-intersections, non-manifoldness). We observe that the contextual and geometric characteristics of 3D printing models differ significantly from those used for computer graphics applications, not to mention standard models (e.g., Stanford bunny, Armadillo, Fertility). We present a new dataset of 10,000 models collected from an online 3D printing model-sharing database. Via analysis of both geometric (e.g., triangle aspect ratios, manifoldness) and contextual (e.g., licenses, tags, classes) characteristics, we demonstrate that this dataset represents a more concise summary of real-world models used for 3D printing compared to existing datasets. To facilitate future research endeavors, we also present an online query interface to select subsets of the dataset according to project-specific characteristics. The complete dataset and per-model statistical data are freely available to the public.

Safety-aligned language models refuse harmful requests through learned refusal behaviors encoded in their internal representations. Recent activation-based jailbreaking methods circumvent these safety mechanisms by applying orthogonal projections to remove refusal directions, but these approaches treat refusal as a one-dimensional phenomenon and ignore the rich distributional structure of model activations. We introduce a principled framework based on optimal transport theory that transforms the entire distribution of harmful activations to match harmless ones. By combining PCA with closed-form Gaussian optimal transport, we achieve efficient computation in high-dimensional representation spaces while preserving essential geometric structure. Across six models (Llama-2, Llama-3.1, Qwen-2.5; 7B-32B parameters), our method achieves up to 11% higher attack success rates than state-of-the-art baselines while maintaining comparable perplexity, demonstrating superior preservation of model capabilities. Critically, we discover that layer-selective intervention (applying optimal transport to 1-2 carefully chosen layers at approximately 40-60% network depth) substantially outperforms full-network interventions, revealing that refusal mechanisms may be localized rather than distributed. Our analysis provides new insights into the geometric structure of safety representations and suggests that current alignment methods may be vulnerable to distributional attacks beyond simple direction removal.

Training large language models (LLMs) relies almost exclusively on dense adaptive optimizers with increasingly sophisticated preconditioners. We challenge this by showing that randomly masking parameter updates can be highly effective, with a masked variant of RMSProp consistently outperforming recent state-of-the-art optimizers. Our analysis reveals that the random masking induces a curvature-dependent geometric regularization that smooths the optimization trajectory. Motivated by this finding, we introduce Momentum-aligned gradient masking (Magma), which modulates the masked updates using momentum-gradient alignment. Extensive LLM pre-training experiments show that Magma is a simple drop-in replacement for adaptive optimizers with consistent gains and negligible computational overhead. Notably, for the 1B model size, Magma reduces perplexity by over 19\% and 9\% compared to Adam and Muon, respectively.

Vision-language agents have achieved remarkable progress in a variety of multimodal reasoning tasks; however, their learning remains constrained by the limitations of human-annotated supervision. Recent self-rewarding approaches attempt to overcome this constraint by allowing models to act as their own critics or reward providers. Yet, purely text-based self-evaluation struggles to verify complex visual reasoning steps and often suffers from evaluation hallucinations. To address these challenges, inspired by recent advances in tool-integrated reasoning, we propose Agent0-VL, a self-evolving vision-language agent that achieves continual improvement with tool-integrated reasoning. Agent0-VL incorporates tool usage not only into reasoning but also into self-evaluation and self-repair, enabling the model to introspect, verify, and refine its reasoning through evidence-grounded analysis. It unifies two synergistic roles within a single LVLM: a Solver that performs multi-turn tool-integrated reasoning, and a Verifier that generates structured feedback and fine-grained self-rewards through tool-grounded critique. These roles interact through a Self-Evolving Reasoning Cycle, where tool-based verification and reinforcement learning jointly align the reasoning and evaluation distributions for stable self-improvement. Through this zero-external-reward evolution, Agent0-VL aligns its reasoning and verification behaviors without any human annotation or external reward models, achieving continual self-improvement. Experiments on geometric problem solving and visual scientific analysis show that Agent0-VL achieves an 12.5% improvement over the base model. Our code is available at https://github.com/aiming-lab/Agent0/Agent0-VL{this https URL}.

Self-supervised learning has emerged as a promising approach for acquiring transferable 3D representations from unlabeled 3D point clouds. Unlike 2D images, which are widely accessible, acquiring 3D assets requires specialized expertise or professional 3D scanning equipment, making it difficult to scale and raising copyright concerns. To address these challenges, we propose learning 3D representations from procedural 3D programs that automatically generate 3D shapes using simple primitives and augmentations. Remarkably, despite lacking semantic content, the 3D representations learned from this synthesized dataset perform on par with state-of-the-art representations learned from semantically recognizable 3D models (e.g., airplanes) across various downstream 3D tasks, including shape classification, part segmentation, and masked point cloud completion. Our analysis further suggests that current self-supervised learning methods primarily capture geometric structures rather than high-level semantics.

Contrastive audio-language pretraining yields powerful joint representations, yet a persistent audio-text modality gap limits the benefits of coupling multimodal encoders with large language models (LLMs). We present Diffusion-Link, a diffusion-based modality-bridging module that generatively maps audio embeddings into the text-embedding distribution. The module is trained at the output embedding from the frozen multimodal encoder and implemented as a lightweight network with three residual MLP blocks. To assess the effect of Diffusion-Link on multimodal encoder-LLM coupling, we evaluate on Automatic Audio Captioning (AAC); to our knowledge, this is the first application of diffusion-based modality bridging to AAC. We report two results. (1) Modality-gap analysis: on similarity and geometric criteria, Diffusion-Link reduces the modality gap the most among prior diffusion-based methods and shows a collective migration of audio embeddings toward the text distribution. (2) Downstream AAC: attaching Diffusion-Link to the same multimodal LLM baseline achieves state-of-the-art on AudioCaps in both zero-shot and fully supervised captioning without external knowledge, with relative gains up to 52.5% and 7.5%, respectively. These findings show that closing the modality gap is pivotal for effective coupling between multimodal encoders and LLMs, and diffusion-based modality bridging offers a promising direction beyond knowledge-retrieval-centric designs. Code will be released upon acceptance https://github.com/DevKiHyun/Diffusion-Link

Sonification -- the mapping of data to non-speech audio -- offers an underexplored channel for representing complex dynamical systems. We treat El Niño-Southern Oscillation (ENSO), a canonical example of low-dimensional climate chaos, as a test case for culturally-situated sonification evaluated through complex systems diagnostics. Using parameter-mapping sonification of the Niño 3.4 sea surface temperature anomaly index (1870--2024), we encode ENSO variability into two traditional Javanese gamelan pentatonic systems (pelog and slendro) across four composition strategies, then analyze the resulting audio as trajectories in a two-dimensional acoustic phase space. Recurrence-based diagnostics, convex hull geometry, and coupling analysis reveal that the sonification pipeline preserves key dynamical signatures: alternating modes produce the highest trajectory recurrence rates, echoing ENSO's quasi-periodicity; layered polyphonic modes explore the broadest phase space regions; and the two scale families induce qualitatively distinct coupling regimes between spectral brightness and energy -- predominantly anti-phase in pelog but near-independent in slendro. Phase space trajectory analysis provides a rigorous geometric framework for comparing sonification designs within a complex systems context. Perceptual validation remains necessary; we contribute the dynamical systems methodology for evaluating such mappings.

Spatial understanding over continuous visual input is crucial for MLLMs to evolve into general-purpose assistants in physical environments. Yet there is still no comprehensive benchmark that holistically assesses the progress toward this goal. In this work, we introduce MMSI-Video-Bench, a fully human-annotated benchmark for video-based spatial intelligence in MLLMs. It operationalizes a four-level framework, Perception, Planning, Prediction, and Cross-Video Reasoning, through 1,106 questions grounded in 1,278 clips from 25 datasets and in-house videos. Each item is carefully designed and reviewed by 3DV experts with explanatory rationales to ensure precise, unambiguous grounding. Leveraging its diverse data sources and holistic task coverage, MMSI-Video-Bench also supports three domain-oriented sub-benchmarks (Indoor Scene Perception Bench, Robot Bench and Grounding Bench) for targeted capability assessment. We evaluate 25 strong open-source and proprietary MLLMs, revealing a striking human--AI gap: many models perform near chance, and the best reasoning model lags humans by nearly 60%. We further find that spatially fine-tuned models still fail to generalize effectively on our benchmark. Fine-grained error analysis exposes systematic failures in geometric reasoning, motion grounding, long-horizon prediction, and cross-video correspondence. We also show that typical frame-sampling strategies transfer poorly to our reasoning-intensive benchmark, and that neither 3D spatial cues nor chain-of-thought prompting yields meaningful gains. We expect our benchmark to establish a solid testbed for advancing video-based spatial intelligence.

Humans can learn to manipulate new objects by simply watching others; providing robots with the ability to learn from such demonstrations would enable a natural interface specifying new behaviors. This work develops Robot See Robot Do (RSRD), a method for imitating articulated object manipulation from a single monocular RGB human demonstration given a single static multi-view object scan. We first propose 4D Differentiable Part Models (4D-DPM), a method for recovering 3D part motion from a monocular video with differentiable rendering. This analysis-by-synthesis approach uses part-centric feature fields in an iterative optimization which enables the use of geometric regularizers to recover 3D motions from only a single video. Given this 4D reconstruction, the robot replicates object trajectories by planning bimanual arm motions that induce the demonstrated object part motion. By representing demonstrations as part-centric trajectories, RSRD focuses on replicating the demonstration's intended behavior while considering the robot's own morphological limits, rather than attempting to reproduce the hand's motion. We evaluate 4D-DPM's 3D tracking accuracy on ground truth annotated 3D part trajectories and RSRD's physical execution performance on 9 objects across 10 trials each on a bimanual YuMi robot. Each phase of RSRD achieves an average of 87% success rate, for a total end-to-end success rate of 60% across 90 trials. Notably, this is accomplished using only feature fields distilled from large pretrained vision models -- without any task-specific training, fine-tuning, dataset collection, or annotation. Project page: https://robot-see-robot-do.github.io

Deep generative neural networks, such as Variational AutoEncoders (VAEs), offer an opportunity to better understand and control language models from the perspective of sentence-level latent spaces. To combine the controllability of VAE latent spaces with the state-of-the-art performance of recent large language models (LLMs), we present in this work LlaMaVAE, which combines expressive encoder and decoder models (sentenceT5 and LlaMA) with a VAE architecture, aiming to provide better text generation control to LLMs. In addition, to conditionally guide the VAE generation, we investigate a new approach based on flow-based invertible neural networks (INNs) named Invertible CVAE. Experimental results reveal that LlaMaVAE can outperform the previous state-of-the-art VAE language model, Optimus, across various tasks, including language modelling, semantic textual similarity and definition modelling. Qualitative analysis on interpolation and traversal experiments also indicates an increased degree of semantic clustering and geometric consistency, which enables better generation control.

The role of the attention mechanism in encoding linguistic knowledge has received special interest in NLP. However, the ability of the attention heads to judge the grammatical acceptability of a sentence has been underexplored. This paper approaches the paradigm of acceptability judgments with topological data analysis (TDA), showing that the geometric properties of the attention graph can be efficiently exploited for two standard practices in linguistics: binary judgments and linguistic minimal pairs. Topological features enhance the BERT-based acceptability classifier scores by 8%-24% on CoLA in three languages (English, Italian, and Swedish). By revealing the topological discrepancy between attention maps of minimal pairs, we achieve the human-level performance on the BLiMP benchmark, outperforming nine statistical and Transformer LM baselines. At the same time, TDA provides the foundation for analyzing the linguistic functions of attention heads and interpreting the correspondence between the graph features and grammatical phenomena.

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Videos inherently represent 2D projections of a dynamic 3D world. However, our analysis suggests that video diffusion models trained solely on raw video data often fail to capture meaningful geometric-aware structure in their learned representations. To bridge this gap between video diffusion models and the underlying 3D nature of the physical world, we propose Geometry Forcing, a simple yet effective method that encourages video diffusion models to internalize latent 3D representations. Our key insight is to guide the model's intermediate representations toward geometry-aware structure by aligning them with features from a pretrained geometric foundation model. To this end, we introduce two complementary alignment objectives: Angular Alignment, which enforces directional consistency via cosine similarity, and Scale Alignment, which preserves scale-related information by regressing unnormalized geometric features from normalized diffusion representation. We evaluate Geometry Forcing on both camera view-conditioned and action-conditioned video generation tasks. Experimental results demonstrate that our method substantially improves visual quality and 3D consistency over the baseline methods. Project page: https://GeometryForcing.github.io.

We investigate the ability of Vision Language Models (VLMs) to perform visual perspective taking using a novel set of visual tasks inspired by established human tests. Our approach leverages carefully controlled scenes, in which a single humanoid minifigure is paired with a single object. By systematically varying spatial configurations - such as object position relative to the humanoid minifigure and the humanoid minifigure's orientation - and using both bird's-eye and surface-level views, we created 144 unique visual tasks. Each visual task is paired with a series of 7 diagnostic questions designed to assess three levels of visual cognition: scene understanding, spatial reasoning, and visual perspective taking. Our evaluation of several state-of-the-art models, including GPT-4-Turbo, GPT-4o, Llama-3.2-11B-Vision-Instruct, and variants of Claude Sonnet, reveals that while they excel in scene understanding, the performance declines significantly on spatial reasoning and further deteriorates on perspective-taking. Our analysis suggests a gap between surface-level object recognition and the deeper spatial and perspective reasoning required for complex visual tasks, pointing to the need for integrating explicit geometric representations and tailored training protocols in future VLM development.

Recent works have shown that, when trained at scale, uni-modal 2D vision and text encoders converge to learned features that share remarkable structural properties, despite arising from different representations. However, the role of 3D encoders with respect to other modalities remains unexplored. Furthermore, existing 3D foundation models that leverage large datasets are typically trained with explicit alignment objectives with respect to frozen encoders from other representations. In this work, we investigate the possibility of a posteriori alignment of representations obtained from uni-modal 3D encoders compared to text-based feature spaces. We show that naive post-training feature alignment of uni-modal text and 3D encoders results in limited performance. We then focus on extracting subspaces of the corresponding feature spaces and discover that by projecting learned representations onto well-chosen lower-dimensional subspaces the quality of alignment becomes significantly higher, leading to improved accuracy on matching and retrieval tasks. Our analysis further sheds light on the nature of these shared subspaces, which roughly separate between semantic and geometric data representations. Overall, ours is the first work that helps to establish a baseline for post-training alignment of 3D uni-modal and text feature spaces, and helps to highlight both the shared and unique properties of 3D data compared to other representations.

A language model (LM) is a mapping from a linguistic context to an output token. However, much remains to be known about this mapping, including how its geometric properties relate to its function. We take a high-level geometric approach to its analysis, observing, across five pre-trained transformer-based LMs and three input datasets, a distinct phase characterized by high intrinsic dimensionality. During this phase, representations (1) correspond to the first full linguistic abstraction of the input; (2) are the first to viably transfer to downstream tasks; (3) predict each other across different LMs. Moreover, we find that an earlier onset of the phase strongly predicts better language modelling performance. In short, our results suggest that a central high-dimensionality phase underlies core linguistic processing in many common LM architectures.

Audio-visual zero-shot learning aims to classify samples consisting of a pair of corresponding audio and video sequences from classes that are not present during training. An analysis of the audio-visual data reveals a large degree of hyperbolicity, indicating the potential benefit of using a hyperbolic transformation to achieve curvature-aware geometric learning, with the aim of exploring more complex hierarchical data structures for this task. The proposed approach employs a novel loss function that incorporates cross-modality alignment between video and audio features in the hyperbolic space. Additionally, we explore the use of multiple adaptive curvatures for hyperbolic projections. The experimental results on this very challenging task demonstrate that our proposed hyperbolic approach for zero-shot learning outperforms the SOTA method on three datasets: VGGSound-GZSL, UCF-GZSL, and ActivityNet-GZSL achieving a harmonic mean (HM) improvement of around 3.0%, 7.0%, and 5.3%, respectively.

IR drop analysis is essential in physical chip design to ensure the power integrity of on-chip power delivery networks. Traditional Electronic Design Automation (EDA) tools have become slow and expensive as transistor density scales. Recent works have introduced machine learning (ML)-based methods that formulate IR drop analysis as an image prediction problem. These existing ML approaches fail to capture both local and long-range dependencies and ignore crucial geometrical and topological information from physical layouts and logical connectivity. To address these limitations, we propose GIF, a Generative IR drop Framework that uses both geometrical and topological information to generate IR drop images. GIF fuses image and graph features to guide a conditional diffusion process, producing high-quality IR drop images. For instance, On the CircuitNet-N28 dataset, GIF achieves 0.78 SSIM, 0.95 Pearson correlation, 21.77 PSNR, and 0.026 NMAE, outperforming prior methods. These results demonstrate that our framework, using diffusion based multimodal conditioning, reliably generates high quality IR drop images. This shows that IR drop analysis can effectively leverage recent advances in generative modeling when geometric layout features and logical circuit topology are jointly modeled. By combining geometry aware spatial features with logical graph representations, GIF enables IR drop analysis to benefit from recent advances in generative modeling for structured image generation.

Geometric Deep Learning has recently attracted significant interest in a wide range of machine learning fields, including document analysis. The application of Graph Neural Networks (GNNs) has become crucial in various document-related tasks since they can unravel important structural patterns, fundamental in key information extraction processes. Previous works in the literature propose task-driven models and do not take into account the full power of graphs. We propose Doc2Graph, a task-agnostic document understanding framework based on a GNN model, to solve different tasks given different types of documents. We evaluated our approach on two challenging datasets for key information extraction in form understanding, invoice layout analysis and table detection. Our code is freely accessible on https://github.com/andreagemelli/doc2graph.

Learning unified text embeddings that excel across diverse downstream tasks is a central goal in representation learning, yet negative transfer remains a persistent obstacle. This challenge is particularly pronounced when jointly training a single encoder for Information Retrieval (IR) and Semantic Textual Similarity (STS), two essential but fundamentally disparate tasks for which naive co-training typically yields steep performance trade-offs. We argue that resolving this conflict requires systematically decoupling task-specific learning signals throughout the training pipeline. To this end, we introduce CoDiEmb, a unified framework that reconciles the divergent requirements of IR and STS in a collaborative yet distinct manner. CoDiEmb integrates three key innovations for effective joint optimization: (1) Task-specialized objectives paired with a dynamic sampler that forms single-task batches and balances per-task updates, thereby preventing gradient interference. For IR, we employ a contrastive loss with multiple positives and hard negatives, augmented by cross-device sampling. For STS, we adopt order-aware objectives that directly optimize correlation and ranking consistency. (2) A delta-guided model fusion strategy that computes fine-grained merging weights for checkpoints by analyzing each parameter's deviation from its pre-trained initialization, proving more effective than traditional Model Soups. (3) An efficient, single-stage training pipeline that is simple to implement and converges stably. Extensive experiments on 15 standard IR and STS benchmarks across three base encoders validate CoDiEmb. Our results and analysis demonstrate that the framework not only mitigates cross-task trade-offs but also measurably improves the geometric properties of the embedding space.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Vision-language models encode continuous geometry that their text pathway fails to express: a 6,000-parameter linear probe extracts hand joint angles at 6.1 degrees MAE from frozen features, while the best text output achieves only 20.0 degrees -- a 3.3x bottleneck. LoRA fine-tuning (r=16, 2,000 images) narrows this gap to 6.5 degrees, providing evidence for a pathway-training deficit rather than a representational one. Training objective determines accuracy more than architecture: five encoders spanning self-supervised, contrastive, and hybrid paradigms converge to statistically equivalent accuracy (R^2 approximately 0.55, TOST-equivalent at delta=0.03) despite sharing as little as CKA=0.41 representational similarity -- functional convergence without representational convergence. Autoregressive generation damages geometric fidelity, but the damage originates in the generation process, not in language alignment: Qwen2.5-VL's LLM layers actually improve probe accuracy over its raw vision encoder. Layer-wise analysis reveals a universal mid-network accuracy peak across all architectures, with attention heads in layers 18-22 carrying disproportionate geometric signal. These findings enable a single frozen backbone to function as a multi-task geometric sensor through lightweight probes, without fine-tuning or text generation.

We present SplatFace, a novel Gaussian splatting framework designed for 3D human face reconstruction without reliance on accurate pre-determined geometry. Our method is designed to simultaneously deliver both high-quality novel view rendering and accurate 3D mesh reconstructions. We incorporate a generic 3D Morphable Model (3DMM) to provide a surface geometric structure, making it possible to reconstruct faces with a limited set of input images. We introduce a joint optimization strategy that refines both the Gaussians and the morphable surface through a synergistic non-rigid alignment process. A novel distance metric, splat-to-surface, is proposed to improve alignment by considering both the Gaussian position and covariance. The surface information is also utilized to incorporate a world-space densification process, resulting in superior reconstruction quality. Our experimental analysis demonstrates that the proposed method is competitive with both other Gaussian splatting techniques in novel view synthesis and other 3D reconstruction methods in producing 3D face meshes with high geometric precision.

The efficacy of deep residual networks is fundamentally predicated on the identity shortcut connection. While this mechanism effectively mitigates the vanishing gradient problem, it imposes a strictly additive inductive bias on feature transformations, thereby limiting the network's capacity to model complex state transitions. In this paper, we introduce Deep Delta Learning (DDL), a novel architecture that generalizes the standard residual connection by modulating the identity shortcut with a learnable, data-dependent geometric transformation. This transformation, termed the Delta Operator, constitutes a rank-1 perturbation of the identity matrix, parameterized by a reflection direction vector k(X) and a gating scalar β(X). We provide a spectral analysis of this operator, demonstrating that the gate β(X) enables dynamic interpolation between identity mapping, orthogonal projection, and geometric reflection. Furthermore, we restructure the residual update as a synchronous rank-1 injection, where the gate acts as a dynamic step size governing both the erasure of old information and the writing of new features. This unification empowers the network to explicitly control the spectrum of its layer-wise transition operator, enabling the modeling of complex, non-monotonic dynamics while preserving the stable training characteristics of gated residual architectures.

Promptable segmentation foundation models such as SAM3 have demonstrated strong generalization capabilities through interactive and concept-based prompting. However, their direct applicability to medical image segmentation remains limited by severe domain shifts, the absence of privileged spatial prompts, and the need to reason over complex anatomical and volumetric structures. Here we present Medical SAM3, a foundation model for universal prompt-driven medical image segmentation, obtained by fully fine-tuning SAM3 on large-scale, heterogeneous 2D and 3D medical imaging datasets with paired segmentation masks and text prompts. Through a systematic analysis of vanilla SAM3, we observe that its performance degrades substantially on medical data, with its apparent competitiveness largely relying on strong geometric priors such as ground-truth-derived bounding boxes. These findings motivate full model adaptation beyond prompt engineering alone. By fine-tuning SAM3's model parameters on 33 datasets spanning 10 medical imaging modalities, Medical SAM3 acquires robust domain-specific representations while preserving prompt-driven flexibility. Extensive experiments across organs, imaging modalities, and dimensionalities demonstrate consistent and significant performance gains, particularly in challenging scenarios characterized by semantic ambiguity, complex morphology, and long-range 3D context. Our results establish Medical SAM3 as a universal, text-guided segmentation foundation model for medical imaging and highlight the importance of holistic model adaptation for achieving robust prompt-driven segmentation under severe domain shift. Code and model will be made available at https://github.com/AIM-Research-Lab/Medical-SAM3.

Vision Language Models (VLMs) exhibit a fundamental semantic-to-geometric gap in spatial reasoning: they excel at qualitative semantic inference but their reasoning operates within a lossy semantic space, misaligned with high-fidelity geometry. Current paradigms fail to bridge this gap. Training-based methods suffer from an ``oracle paradox,'' learning flawed spatial logic from imperfect oracles. Tool-integrated methods constrain the final computation but critically leave the VLM's planning process unconstrained, resulting in geometrically flawed plans. In this work, we propose Geometrically-Constrained Agent (GCA), a training-free agentic paradigm that resolves this gap by introducing a formal task constraint. Specifically, we strategically decouples the VLM's role into two stages. First, acting as a semantic analyst, the VLM translates the user's ambiguous query into the formal, verifiable task constraint, which defines the reference frame and objective. Second, acting as a task solver, the VLM generates and executes tool calls strictly within the deterministic bounds defined by the constraint. This geometrically-constrained reasoning strategy successfully resolve the semantic-to-geometric gap, yielding a robust and verifiable reasoning pathway for spatial reasoning. Comprehensive experiments demonstrate that GCA achieves SOTA performance on multiple spatial reasoning benchmarks, surpassing existing training-based and tool-integrated methods by ~27%. Please see our homepage at https://gca-spatial-reasoning.github.io.

In the realm of text-to-3D generation, utilizing 2D diffusion models through score distillation sampling (SDS) frequently leads to issues such as blurred appearances and multi-faced geometry, primarily due to the intrinsically noisy nature of the SDS loss. Our analysis identifies the core of these challenges as the interaction among noise levels in the 2D diffusion process, the architecture of the diffusion network, and the 3D model representation. To overcome these limitations, we present StableDreamer, a methodology incorporating three advances. First, inspired by InstructNeRF2NeRF, we formalize the equivalence of the SDS generative prior and a simple supervised L2 reconstruction loss. This finding provides a novel tool to debug SDS, which we use to show the impact of time-annealing noise levels on reducing multi-faced geometries. Second, our analysis shows that while image-space diffusion contributes to geometric precision, latent-space diffusion is crucial for vivid color rendition. Based on this observation, StableDreamer introduces a two-stage training strategy that effectively combines these aspects, resulting in high-fidelity 3D models. Third, we adopt an anisotropic 3D Gaussians representation, replacing Neural Radiance Fields (NeRFs), to enhance the overall quality, reduce memory usage during training, and accelerate rendering speeds, and better capture semi-transparent objects. StableDreamer reduces multi-face geometries, generates fine details, and converges stably.

We study how large language models (LLMs) ``think'' through their representation space. We propose a novel geometric framework that models an LLM's reasoning as flows -- embedding trajectories evolving where logic goes. We disentangle logical structure from semantics by employing the same natural deduction propositions with varied semantic carriers, allowing us to test whether LLMs internalize logic beyond surface form. This perspective connects reasoning with geometric quantities such as position, velocity, and curvature, enabling formal analysis in representation and concept spaces. Our theory establishes: (1) LLM reasoning corresponds to smooth flows in representation space, and (2) logical statements act as local controllers of these flows' velocities. Using learned representation proxies, we design controlled experiments to visualize and quantify reasoning flows, providing empirical validation of our theoretical framework. Our work serves as both a conceptual foundation and practical tools for studying reasoning phenomenon, offering a new lens for interpretability and formal analysis of LLMs' behavior.

This paper investigates the fundamental discontinuity between the latest two Segment Anything Models: SAM2 and SAM3. We explain why the expertise in prompt-based segmentation of SAM2 does not transfer to the multimodal concept-driven paradigm of SAM3. SAM2 operates through spatial prompts points, boxes, and masks yielding purely geometric and temporal segmentation. In contrast, SAM3 introduces a unified vision-language architecture capable of open-vocabulary reasoning, semantic grounding, contrastive alignment, and exemplar-based concept understanding. We structure this analysis through five core components: (1) a Conceptual Break Between Prompt-Based and Concept-Based Segmentation, contrasting spatial prompt semantics of SAM2 with multimodal fusion and text-conditioned mask generation of SAM3; (2) Architectural Divergence, detailing pure vision-temporal design of SAM2 versus integration of vision-language encoders, geometry and exemplar encoders, fusion modules, DETR-style decoders, object queries, and ambiguity-handling via Mixture-of-Experts in SAM3; (3) Dataset and Annotation Differences, contrasting SA-V video masks with multimodal concept-annotated corpora of SAM3; (4) Training and Hyperparameter Distinctions, showing why SAM2 optimization knowledge does not apply to SAM3; and (5) Evaluation, Metrics, and Failure Modes, outlining the transition from geometric IoU metrics to semantic, open-vocabulary evaluation. Together, these analyses establish SAM3 as a new class of segmentation foundation model and chart future directions for the emerging concept-driven segmentation era.

Molecule generation is advancing rapidly in chemical discovery and drug design. Flow matching methods have recently set the state of the art (SOTA) in unconditional molecule generation, surpassing score-based diffusion models. However, diffusion models still lead in property-guided generation. In this work, we introduce PropMolFlow, a novel approach for property-guided molecule generation based on geometry-complete SE(3)-equivariant flow matching. Integrating five different property embedding methods with a Gaussian expansion of scalar properties, PropMolFlow outperforms previous SOTA diffusion models in conditional molecule generation across various properties while preserving the stability and validity of the generated molecules, consistent with its unconditional counterpart. Additionally, it enables faster inference with significantly fewer time steps compared to baseline models. We highlight the importance of validating the properties of generated molecules through DFT calculations performed at the same level of theory as the training data. Specifically, our analysis identifies properties that require DFT validation and others where a pretrained SE(3) geometric vector perceptron regressors provide sufficiently accurate predictions on generated molecules. Furthermore, we introduce a new property metric designed to assess the model's ability to propose molecules with underrepresented property values, assessing its capacity for out-of-distribution generalization. Our findings reveal shortcomings in existing structural metrics, which mistakenly validate open-shell molecules or molecules with invalid valence-charge configurations, underscoring the need for improved evaluation frameworks. Overall, this work paves the way for developing targeted property-guided generation methods, enhancing the design of molecular generative models for diverse applications.

Multimodal foundation models, such as GPT-4o, have recently made remarkable progress, but it is not clear where exactly these models stand in terms of understanding vision. In this paper, we benchmark the performance of popular multimodal foundation models (GPT-4o, o4-mini, Gemini 1.5 Pro and Gemini 2.0 Flash, Claude 3.5 Sonnet, Qwen2-VL, Llama 3.2) on standard computer vision tasks (semantic segmentation, object detection, image classification, depth and surface normal prediction) using established datasets (e.g., COCO, ImageNet and its variants, etc). The main challenges to performing this are: 1) most models are trained to output text and cannot natively express versatile domains, such as segments or 3D geometry, and 2) many leading models are proprietary and accessible only at an API level, i.e., there is no weight access to adapt them. We address these challenges by translating standard vision tasks into equivalent text-promptable and API-compatible tasks via prompt chaining to create a standardized benchmarking framework. We observe that 1) the models are not close to the state-of-the-art specialist models at any task. However, 2) they are respectable generalists; this is remarkable as they are presumably trained on primarily image-text-based tasks. 3) They perform semantic tasks notably better than geometric ones. 4) While the prompt-chaining techniques affect performance, better models exhibit less sensitivity to prompt variations. 5) GPT-4o performs the best among non-reasoning models, securing the top position in 4 out of 6 tasks, 6) reasoning models, e.g. o3, show improvements in geometric tasks, and 7) a preliminary analysis of models with native image generation, like the latest GPT-4o, shows they exhibit quirks like hallucinations and spatial misalignments.

3D vector graphics play a crucial role in various applications including 3D shape retrieval, conceptual design, and virtual reality interactions due to their ability to capture essential structural information with minimal representation. While recent approaches have shown promise in generating 3D vector graphics, they often suffer from lengthy processing times and struggle to maintain view consistency. To address these limitations, we propose ViewCraft3D (VC3D), an efficient method that leverages 3D priors to generate 3D vector graphics. Specifically, our approach begins with 3D object analysis, employs a geometric extraction algorithm to fit 3D vector graphics to the underlying structure, and applies view-consistent refinement process to enhance visual quality. Our comprehensive experiments demonstrate that VC3D outperforms previous methods in both qualitative and quantitative evaluations, while significantly reducing computational overhead. The resulting 3D sketches maintain view consistency and effectively capture the essential characteristics of the original objects.

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum-method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations.

A core aspect of human perception is situated awareness, the ability to relate ourselves to the surrounding physical environment and reason over possible actions in context. However, most existing benchmarks for multimodal foundation models (MFMs) emphasize environment-centric spatial relations (relations among objects in a scene), while largely overlooking observer-centric relationships that require reasoning relative to agent's viewpoint, pose, and motion. To bridge this gap, we introduce SAW-Bench (Situated Awareness in the Real World), a novel benchmark for evaluating egocentric situated awareness using real-world videos. SAW-Bench comprises 786 self-recorded videos captured with Ray-Ban Meta (Gen 2) smart glasses spanning diverse indoor and outdoor environments, and over 2,071 human-annotated question-answer pairs. It probes a model's observer-centric understanding with six different awareness tasks. Our comprehensive evaluation reveals a human-model performance gap of 37.66%, even with the best-performing MFM, Gemini 3 Flash. Beyond this gap, our in-depth analysis uncovers several notable findings; for example, while models can exploit partial geometric cues in egocentric videos, they often fail to infer a coherent camera geometry, leading to systematic spatial reasoning errors. We position SAW-Bench as a benchmark for situated spatial intelligence, moving beyond passive observation to understanding physically grounded, observer-centric dynamics.

Large Vision-Language Models (LVLMs) hold great promise for advancing remote sensing (RS) analysis, yet existing reasoning segmentation frameworks couple linguistic reasoning and pixel prediction through end-to-end supervised fine-tuning, leading to weak geometric grounding and limited generalization across tasks. To address this, we developed Think2Seg-RS, a decoupled framework that trains an LVLM prompter to control a frozen Segment Anything Model (SAM) via structured geometric prompts. Through a mask-only reinforcement learning objective, the LVLM learns to translate abstract semantic reasoning into spatially grounded actions, achieving state-of-the-art performance on the EarthReason dataset. Remarkably, the learned prompting policy generalizes zero-shot to multiple referring segmentation benchmarks, exposing a distinct divide between semantic-level and instance-level grounding. We further found that compact segmenters outperform larger ones under semantic-level supervision, and that negative prompts are ineffective in heterogeneous aerial backgrounds. Together, these findings establish semantic-level reasoning segmentation as a new paradigm for geospatial understanding, opening the way toward unified, interpretable LVLM-driven Earth observation. Our code and model are available at https://github.com/Ricardo-XZ/Think2Seg-RS.

We study a persistent failure mode in multi-objective alignment for large language models (LLMs): training improves performance on only a subset of objectives while causing others to degrade. We formalize this phenomenon as cross-objective interference and conduct the first systematic study across classic scalarization algorithms, showing that interference is pervasive and exhibits strong model dependence. To explain this phenomenon, we derive a local covariance law showing that an objective improves at first order when its reward exhibits positive covariance with the scalarized score. We extend this analysis to clipped surrogate objectives used in modern alignment, demonstrating that the covariance law remains valid under mild conditions despite clipping. Building on this analysis, we propose Covariance Targeted Weight Adaptation (CTWA), a plug-and-play method that maintains positive covariance between objective rewards and the training signal to effectively mitigate cross-objective interference. Finally, we complement these local improvement conditions with a global convergence analysis under the Polyak--Łojasiewicz condition, establishing when non-convex scalarized optimization achieves global convergence and how cross-objective interference depends on specific model geometric properties.

Engineering construction automation aims to transform natural language specifications into physically viable structures, requiring complex integrated reasoning under strict physical constraints. While modern LLMs possess broad knowledge and strong reasoning capabilities that make them promising candidates for this domain, their construction competencies remain largely unevaluated. To address this gap, we introduce BuildArena, the first physics-aligned interactive benchmark designed for language-driven engineering construction. It contributes to the community in four aspects: (1) a highly customizable benchmarking framework for in-depth comparison and analysis of LLMs; (2) an extendable task design strategy spanning static and dynamic mechanics across multiple difficulty tiers; (3) a 3D Spatial Geometric Computation Library for supporting construction based on language instructions; (4) a baseline LLM agentic workflow that effectively evaluates diverse model capabilities. On eight frontier LLMs, BuildArena comprehensively evaluates their capabilities for language-driven and physics-grounded construction automation. The project page is at https://build-arena.github.io/.

In the realm of computer vision and graphics, accurately establishing correspondences between geometric 3D shapes is pivotal for applications like object tracking, registration, texture transfer, and statistical shape analysis. Moving beyond traditional hand-crafted and data-driven feature learning methods, we incorporate spectral methods with deep learning, focusing on functional maps (FMs) and optimal transport (OT). Traditional OT-based approaches, often reliant on entropy regularization OT in learning-based framework, face computational challenges due to their quadratic cost. Our key contribution is to employ the sliced Wasserstein distance (SWD) for OT, which is a valid fast optimal transport metric in an unsupervised shape matching framework. This unsupervised framework integrates functional map regularizers with a novel OT-based loss derived from SWD, enhancing feature alignment between shapes treated as discrete probability measures. We also introduce an adaptive refinement process utilizing entropy regularized OT, further refining feature alignments for accurate point-to-point correspondences. Our method demonstrates superior performance in non-rigid shape matching, including near-isometric and non-isometric scenarios, and excels in downstream tasks like segmentation transfer. The empirical results on diverse datasets highlight our framework's effectiveness and generalization capabilities, setting new standards in non-rigid shape matching with efficient OT metrics and an adaptive refinement module.

Maintaining spatial world consistency over long horizons remains a central challenge for camera-controllable video generation. Existing memory-based approaches often condition generation on globally reconstructed 3D scenes by rendering anchor videos from the reconstructed geometry in the history. However, reconstructing a global 3D scene from multiple views inevitably introduces cross-view misalignment, as pose and depth estimation errors cause the same surfaces to be reconstructed at slightly different 3D locations across views. When fused, these inconsistencies accumulate into noisy geometry that contaminates the conditioning signals and degrades generation quality. We introduce AnchorWeave, a memory-augmented video generation framework that replaces a single misaligned global memory with multiple clean local geometric memories and learns to reconcile their cross-view inconsistencies. To this end, AnchorWeave performs coverage-driven local memory retrieval aligned with the target trajectory and integrates the selected local memories through a multi-anchor weaving controller during generation. Extensive experiments demonstrate that AnchorWeave significantly improves long-term scene consistency while maintaining strong visual quality, with ablation and analysis studies further validating the effectiveness of local geometric conditioning, multi-anchor control, and coverage-driven retrieval.

Precise determination of thermodynamic parameters in ultracold Bose gases remains challenging due to the destructive nature of conventional measurement techniques and inherent experimental uncertainties. We demonstrate an artificial intelligence approach for rapid, non-destructive estimation of the chemical potential and temperature from single-shot, in situ imaged density profiles of finite-temperature Bose gases. Our convolutional neural network is trained exclusively on quasi-2D `pancake' condensates in harmonic trap configurations. It achieves parameter extraction within fractions of a second. The model also demonstrates zero-shot generalisation across both trap geometry and thermalisation dynamics, successfully estimating thermodynamic parameters for toroidally trapped condensates with errors of only a few nanokelvin despite no prior exposure to such geometries during training, and maintaining predictive accuracy during dynamic thermalisation processes after a relatively brief evolution without explicit training on non-equilibrium states. These results suggest that supervised learning can overcome traditional limitations in ultracold atom thermometry, with extension to broader geometric configurations, temperature ranges, and additional parameters potentially enabling comprehensive real-time analysis of quantum gas experiments. Such capabilities could significantly streamline experimental workflows whilst improving measurement precision across a range of quantum fluid systems.

We present ReXVQA, the largest and most comprehensive benchmark for visual question answering (VQA) in chest radiology, comprising approximately 696,000 questions paired with 160,000 chest X-rays studies across training, validation, and test sets. Unlike prior efforts that rely heavily on template based queries, ReXVQA introduces a diverse and clinically authentic task suite reflecting five core radiological reasoning skills: presence assessment, location analysis, negation detection, differential diagnosis, and geometric reasoning. We evaluate eight state-of-the-art multimodal large language models, including MedGemma-4B-it, Qwen2.5-VL, Janus-Pro-7B, and Eagle2-9B. The best-performing model (MedGemma) achieves 83.24% overall accuracy. To bridge the gap between AI performance and clinical expertise, we conducted a comprehensive human reader study involving 3 radiology residents on 200 randomly sampled cases. Our evaluation demonstrates that MedGemma achieved superior performance (83.84% accuracy) compared to human readers (best radiology resident: 77.27%), representing a significant milestone where AI performance exceeds expert human evaluation on chest X-ray interpretation. The reader study reveals distinct performance patterns between AI models and human experts, with strong inter-reader agreement among radiologists while showing more variable agreement patterns between human readers and AI models. ReXVQA establishes a new standard for evaluating generalist radiological AI systems, offering public leaderboards, fine-grained evaluation splits, structured explanations, and category-level breakdowns. This benchmark lays the foundation for next-generation AI systems capable of mimicking expert-level clinical reasoning beyond narrow pathology classification. Our dataset will be open-sourced at https://huggingface.co/datasets/rajpurkarlab/ReXVQA

The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks and Boltzmann machines are versions of the Ising model, a model extensively studied in statistical mechanics for over a century. In the first part of this chapter, we provide an overview of the principles, models, and applications of ANNs, highlighting their connections to statistical mechanics and statistical learning theory. Artificial neural networks can be seen as high-dimensional mathematical functions, and understanding the geometric properties of their loss landscapes (i.e., the high-dimensional space on which one wishes to find extrema or saddles) can provide valuable insights into their optimization behavior, generalization abilities, and overall performance. Visualizing these functions can help us design better optimization methods and improve their generalization abilities. Thus, the second part of this chapter focuses on quantifying geometric properties and visualizing loss functions associated with deep ANNs.

Priorities in multi-criteria decision-making (MCDM) convey the relevance preference of one criterion over another, which is usually reflected by imposing the non-negativity and unit-sum constraints. The processing of such priorities is different than other unconstrained data, but this point is often neglected by researchers, which results in fallacious statistical analysis. This article studies three prevalent fallacies in group MCDM along with solutions based on compositional data analysis to avoid misusing statistical operations. First, we use a compositional approach to aggregate the priorities of a group of DMs and show that the outcome of the compositional analysis is identical to the normalized geometric mean, meaning that the arithmetic mean should be avoided. Furthermore, a new aggregation method is developed, which is a robust surrogate for the geometric mean. We also discuss the errors in computing measures of dispersion, including standard deviation and distance functions. Discussing the fallacies in computing the standard deviation, we provide a probabilistic criteria ranking by developing proper Bayesian tests, where we calculate the extent to which a criterion is more important than another. Finally, we explain the errors in computing the distance between priorities, and a clustering algorithm is specially tailored based on proper distance metrics.

Image spatial editing performs geometry-driven transformations, allowing precise control over object layout and camera viewpoints. Current models are insufficient for fine-grained spatial manipulations, motivating a dedicated assessment suite. Our contributions are listed: (i) We introduce SpatialEdit-Bench, a complete benchmark that evaluates spatial editing by jointly measuring perceptual plausibility and geometric fidelity via viewpoint reconstruction and framing analysis. (ii) To address the data bottleneck for scalable training, we construct SpatialEdit-500k, a synthetic dataset generated with a controllable Blender pipeline that renders objects across diverse backgrounds and systematic camera trajectories, providing precise ground-truth transformations for both object- and camera-centric operations. (iii) Building on this data, we develop SpatialEdit-16B, a baseline model for fine-grained spatial editing. Our method achieves competitive performance on general editing while substantially outperforming prior methods on spatial manipulation tasks. All resources will be made public at https://github.com/EasonXiao-888/SpatialEdit.

Large Language Models (LLMs) display striking surface fluency yet systematically fail at tasks requiring symbolic reasoning, arithmetic accuracy, and logical consistency. This paper offers a structural diagnosis of such failures, revealing a persistent gap between comprehension and competence. Through controlled experiments and architectural analysis, we demonstrate that LLMs often articulate correct principles without reliably applying them--a failure rooted not in knowledge access, but in computational execution. We term this phenomenon the computational split-brain syndrome, where instruction and action pathways are geometrically and functionally dissociated. This core limitation recurs across domains, from mathematical operations to relational inferences, and explains why model behavior remains brittle even under idealized prompting. We argue that LLMs function as powerful pattern completion engines, but lack the architectural scaffolding for principled, compositional reasoning. Our findings delineate the boundary of current LLM capabilities and motivate future models with metacognitive control, principle lifting, and structurally grounded execution. This diagnosis also clarifies why mechanistic interpretability findings may reflect training-specific pattern coordination rather than universal computational principles, and why the geometric separation between instruction and execution pathways suggests limitations in neural introspection and mechanistic analysis.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since most geometric proofs rely on intuitive diagrams, verifying geometry problems is particularly challenging. To address these gaps, we introduce LeanGeo, a unified formal system for formalizing and solving competition-level geometry problems within the Lean 4 theorem prover. LeanGeo features a comprehensive library of high-level geometric theorems with Lean's foundational logic, enabling rigorous proof verification and seamless integration with Mathlib. We also present LeanGeo-Bench, a formal geometry benchmark in LeanGeo, comprising problems from the International Mathematical Olympiad (IMO) and other advanced sources. Our evaluation demonstrates the capabilities and limitations of state-of-the-art Large Language Models on this benchmark, highlighting the need for further advancements in automated geometric reasoning. We open source the theorem library and the benchmark of LeanGeo at https://github.com/project-numina/LeanGeo/tree/master.

Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and proving, especially in geometry for its combination of numerical and spatial elements. We introduce TongGeometry, a Euclidean geometry system supporting tree-search-based guided problem proposing and solving. The efficient geometry system establishes the most extensive repository of geometry theorems to date: within the same computational budget as the existing state-of-the-art, TongGeometry discovers 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion exhibiting geometric symmetry. Among them, 10 theorems were proposed to regional mathematical olympiads with 3 of TongGeometry's proposals selected in real competitions, earning spots in a national team qualifying exam or a top civil olympiad in China and the US. Guided by fine-tuned large language models, TongGeometry solved all International Mathematical Olympiad geometry in IMO-AG-30, outperforming gold medalists for the first time. It also surpasses the existing state-of-the-art across a broader spectrum of olympiad-level problems. The full capabilities of the system can be utilized on a consumer-grade machine, making the model more accessible and fostering widespread democratization of its use. By analogy, unlike existing systems that merely solve problems like students, TongGeometry acts like a geometry coach, discovering, presenting, and proving theorems.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

In this article, we develop an asymptotic method for constructing confidence regions for the set of all linear subspaces arising from PCA, from which we derive hypothesis tests on this set. Our method is based on the geometry of Riemannian manifolds with which some sets of linear subspaces are endowed.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

We analyze surface patches with a corner that is rounded in the sense that the partial derivatives at that point are antiparallel. Sufficient conditions for G^1 smoothness are given, which, up to a certain degenerate case, are also necessary. Further, we investigate curvature integrability and present examples

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly available tool to manipulate knot diagrams in a real-time, interactive way is yet to be developed. We introduce a method of operating on the geometry of the knot diagram itself without any underlying three-dimensional structure that can underpin such an application. This allows us to directly interact with vector graphics knot diagrams while at the same time computing knot invariants in ways proposed by previous work. An implementation of this method is provided.

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types F_4, E_6, E_7, E_8 and G_2, that form a line using the opposition relation. With that, we obtain a classification of so-called ``geometric lines'' in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

In machine learning, there is a long history of trying to build neural networks that can learn from fewer example data by baking in strong geometric priors. However, it is not always clear a priori what geometric constraints are appropriate for a given task. Here, we explore the possibility that one can uncover useful geometric inductive biases by studying how training molds the Riemannian geometry induced by unconstrained neural network feature maps. We first show that at infinite width, neural networks with random parameters induce highly symmetric metrics on input space. This symmetry is broken by feature learning: networks trained to perform classification tasks learn to magnify local areas along decision boundaries. This holds in deep networks trained on high-dimensional image classification tasks, and even in self-supervised representation learning. These results begin to elucidate how training shapes the geometry induced by unconstrained neural network feature maps, laying the groundwork for an understanding of this richly nonlinear form of feature learning.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.