Daily Papers

As Large Language Models (LLMs) have become more advanced, they have outpaced our abilities to accurately evaluate their quality. Not only is finding data to adequately probe particular model properties difficult, but evaluating the correctness of a model's freeform generation alone is a challenge. To address this, many evaluations now rely on using LLMs themselves as judges to score the quality of outputs from other LLMs. Evaluations most commonly use a single large model like GPT4. While this method has grown in popularity, it is costly, has been shown to introduce intramodel bias, and in this work, we find that very large models are often unnecessary. We propose instead to evaluate models using a Panel of LLm evaluators (PoLL). Across three distinct judge settings and spanning six different datasets, we find that using a PoLL composed of a larger number of smaller models outperforms a single large judge, exhibits less intra-model bias due to its composition of disjoint model families, and does so while being over seven times less expensive.

Multi-Agent Systems (MAS) powered by Large Language Models have unlocked advanced collaborative reasoning, yet they remain shackled by the inefficiency of discrete text communication, which imposes significant runtime overhead and information quantization loss. While latent state transfer offers a high-bandwidth alternative, existing approaches either assume homogeneous sender-receiver architectures or rely on pair-specific learned translators, limiting scalability and modularity across diverse model families with disjoint manifolds. In this work, we propose the Vision Wormhole, a novel framework that repurposes the visual interface of Vision-Language Models (VLMs) to enable model-agnostic, text-free communication. By introducing a Universal Visual Codec, we map heterogeneous reasoning traces into a shared continuous latent space and inject them directly into the receiver's visual pathway, effectively treating the vision encoder as a universal port for inter-agent telepathy. Our framework adopts a hub-and-spoke topology to reduce pairwise alignment complexity from O(N^2) to O(N) and leverages a label-free, teacher-student distillation objective to align the high-speed visual channel with the robust reasoning patterns of the text pathway. Extensive experiments across heterogeneous model families (e.g., Qwen-VL, Gemma) demonstrate that the Vision Wormhole reduces end-to-end wall-clock time in controlled comparisons while maintaining reasoning fidelity comparable to standard text-based MAS. Code is available at https://github.com/xz-liu/heterogeneous-latent-mas

Model merging has become one of the key technologies for enhancing the capabilities and efficiency of Large Language Models (LLMs). However, our understanding of the expected performance gains and principles when merging any two models remains limited. In this work, we introduce model kinship, the degree of similarity or relatedness between LLMs, analogous to biological evolution. With comprehensive empirical analysis, we find that there is a certain relationship between model kinship and the performance gains after model merging, which can help guide our selection of candidate models. Inspired by this, we propose a new model merging strategy: Top-k Greedy Merging with Model Kinship, which can yield better performance on benchmark datasets. Specifically, we discover that using model kinship as a criterion can assist us in continuously performing model merging, alleviating the degradation (local optima) in model evolution, whereas model kinship can serve as a guide to escape these traps. Code is available at https://github.com/zjunlp/ModelKinship.

Many have observed that the development and deployment of generative machine learning (ML) and artificial intelligence (AI) models follow a distinctive pattern in which pre-trained models are adapted and fine-tuned for specific downstream tasks. However, there is limited empirical work that examines the structure of these interactions. This paper analyzes 1.86 million models on Hugging Face, a leading peer production platform for model development. Our study of model family trees -- networks that connect fine-tuned models to their base or parent -- reveals sprawling fine-tuning lineages that vary widely in size and structure. Using an evolutionary biology lens to study ML models, we use model metadata and model cards to measure the genetic similarity and mutation of traits over model families. We find that models tend to exhibit a family resemblance, meaning their genetic markers and traits exhibit more overlap when they belong to the same model family. However, these similarities depart in certain ways from standard models of asexual reproduction, because mutations are fast and directed, such that two `sibling' models tend to exhibit more similarity than parent/child pairs. Further analysis of the directional drifts of these mutations reveals qualitative insights about the open machine learning ecosystem: Licenses counter-intuitively drift from restrictive, commercial licenses towards permissive or copyleft licenses, often in violation of upstream license's terms; models evolve from multi-lingual compatibility towards english-only compatibility; and model cards reduce in length and standardize by turning, more often, to templates and automatically generated text. Overall, this work takes a step toward an empirically grounded understanding of model fine-tuning and suggests that ecological models and methods can yield novel scientific insights.

As Large Language Models (LLMs) gain widespread practical application, providing the model family of different parameter sizes has become standard practice to address diverse computational requirements. Conventionally, each model in a family is trained independently, resulting in computational costs that scale additively with the number of models. We propose an efficient method for constructing the model family through progressive training, where smaller models are incrementally expanded to larger sizes to create a complete model family. Through extensive experiments with a model family ranging from 1B to 8B parameters, we demonstrate that our method reduces computational costs by approximately 25% while maintaining comparable performance to independently trained models. Furthermore, by strategically adjusting maximum learning rates based on model size, our method outperforms the independent training across various metrics. Beyond performance gains, our approach offers an additional advantage: models in our family tend to yield more consistent behavior across different model sizes.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

To generalise evolution families we consider systems of contractions {varphi(u, v)}_{(u, v) in E} defined on the edges of a graph G = (Ω, E). In this setup the Markov property, or divisibility, can be modelled via varphi(u, v)varphi(v, w) = varphi(u, w) for edges (u, v), (v, w), (u, w) in E. We obtain results in three settings: 1) contractive Banach space operators; 2) positive unital maps on C^{ast}-algebras; and 3) CPTP-maps on trace class operators on a Hilbert space. In the discrete setting, we are able to dilate possibly indivisible families of contractions to divisible families of operators with 'nice' properties (viz. surjective isometries resp. C^{ast}-algebraic automorphisms resp. unitary representations). In the special case of linearly ordered graphs equipped with the order topology, we establish sufficient conditions for strongly continuous dilations of possibly indivisible families in the Banach space and C^{ast}-algebra contexts. To achieve these results we work with string-rewriting systems, and make use of and extend dilation theorems of Stroescu [44], Kraus [23, 24], and vom Ende--Dirr [50].

The success of text-to-image (T2I) generation models has spurred a proliferation of numerous model checkpoints fine-tuned from the same base model on various specialized datasets. This overwhelming specialized model production introduces new challenges for high parameter redundancy and huge storage cost, thereby necessitating the development of effective methods to consolidate and unify the capabilities of diverse powerful models into a single one. A common practice in model merging adopts static linear interpolation in the parameter space to achieve the goal of style mixing. However, it neglects the features of T2I generation task that numerous distinct models cover sundry styles which may lead to incompatibility and confusion in the merged model. To address this issue, we introduce a style-promptable image generation pipeline which can accurately generate arbitrary-style images under the control of style vectors. Based on this design, we propose the score distillation based model merging paradigm (DMM), compressing multiple models into a single versatile T2I model. Moreover, we rethink and reformulate the model merging task in the context of T2I generation, by presenting new merging goals and evaluation protocols. Our experiments demonstrate that DMM can compactly reorganize the knowledge from multiple teacher models and achieve controllable arbitrary-style generation.

This paper presents a new exploration into a category of diffusion models built upon state space architecture. We endeavor to train diffusion models for image data, wherein the traditional U-Net backbone is supplanted by a state space backbone, functioning on raw patches or latent space. Given its notable efficacy in accommodating long-range dependencies, Diffusion State Space Models (DiS) are distinguished by treating all inputs including time, condition, and noisy image patches as tokens. Our assessment of DiS encompasses both unconditional and class-conditional image generation scenarios, revealing that DiS exhibits comparable, if not superior, performance to CNN-based or Transformer-based U-Net architectures of commensurate size. Furthermore, we analyze the scalability of DiS, gauged by the forward pass complexity quantified in Gflops. DiS models with higher Gflops, achieved through augmentation of depth/width or augmentation of input tokens, consistently demonstrate lower FID. In addition to demonstrating commendable scalability characteristics, DiS-H/2 models in latent space achieve performance levels akin to prior diffusion models on class-conditional ImageNet benchmarks at the resolution of 256times256 and 512times512, while significantly reducing the computational burden. The code and models are available at: https://github.com/feizc/DiS.

For a family of two-matrix models \[ 1{2} Tr(A^2+B^2) - g{4} Tr(A^4+B^4) - cases h{2} Tr( A BA B) \\ h{4} Tr( A BA B+ ABBA ) \\ h{2} Tr( A B BA ) cases \] with hermitian A and B, we provide, in each case, a Monte Carlo estimate of the boundary of the maximal convergence domain in the (h,g)-plane. The results are discussed comparing with exact solutions (in agreement with the only analytically solved case) and phase diagrams obtained by means of the functional renormalization group.

Large language models (LLMs) are typically deployed under diverse memory and compute constraints. Existing approaches build model families by training each size independently, which is prohibitively expensive and provides only coarse-grained size options. In this work, we identify a novel phenomenon that we call boomerang distillation: starting from a large base model (the teacher), one first distills down to a small student and then progressively reconstructs intermediate-sized models by re-incorporating blocks of teacher layers into the student without any additional training. This process produces zero-shot interpolated models of many intermediate sizes whose performance scales smoothly between the student and teacher, often matching or surpassing pretrained or distilled models of the same size. We further analyze when this type of interpolation succeeds, showing that alignment between teacher and student through pruning and distillation is essential. Boomerang distillation thus provides a simple and efficient way to generate fine-grained model families, dramatically reducing training cost while enabling flexible adaptation across deployment environments. The code and models are available at https://github.com/dcml-lab/boomerang-distillation.

As models and data scale, independently trained networks often induce analogous notions of similarity. But, matching similarities is weaker than establishing an explicit correspondence between the representation spaces, especially for multimodal models, where consistency must hold not only within each modality, but also for the learned image-text coupling. We therefore ask: given two independently trained multimodal contrastive models (with encoders (f, g) and (f,g)) -- trained on different distributions and with different architectures -- does a systematic geometric relationship exist between their embedding spaces? If so, what form does it take, and does it hold uniformly across modalities? In this work, we show that across model families such as CLIP, SigLIP, and FLAVA, this geometric relationship is well approximated by an orthogonal map (up to a global mean shift), i.e., there exists an orthogonal map Q where Q^top Q = I such that f(x)approx Q f(x) for paired images x. Strikingly, the same Q simultaneously aligns the text encoders i.e., g(y)approx Q g(y) for texts y. Theoretically, we prove that if the multimodal kernel agrees across models on a small anchor set i.e. langle f(x), g(y)rangle approx langle f(x), g(y)rangle, then the two models must be related by a single orthogonal map Q and the same Q maps images and text across models. More broadly, this finding enables backward-compatible model upgrades, avoiding costly re-embedding, and has implications for the privacy of learned representations. Our project page: https://canonical-multimodal.github.io/

Modern model hubs, such as Hugging Face, store tens of petabytes of LLMs, with fine-tuned variants vastly outnumbering base models and dominating storage consumption. Existing storage reduction techniques -- such as deduplication and compression -- are either LLM-oblivious or not compatible with each other, limiting data reduction effectiveness. Our large-scale characterization study across all publicly available Hugging Face LLM repositories reveals several key insights: (1) fine-tuned models within the same family exhibit highly structured, sparse parameter differences suitable for delta compression; (2) bitwise similarity enables LLM family clustering; and (3) tensor-level deduplication is better aligned with model storage workloads, achieving high data reduction with low metadata overhead. Building on these insights, we design BitX, an effective, fast, lossless delta compression algorithm that compresses XORed difference between fine-tuned and base LLMs. We build ZipLLM, a model storage reduction pipeline that unifies tensor-level deduplication and lossless BitX compression. By synergizing deduplication and compression around LLM family clustering, ZipLLM reduces model storage consumption by 54%, over 20% higher than state-of-the-art deduplication and compression approaches.

The public model zoo containing enormous powerful pretrained model families (e.g., ResNet/DeiT) has reached an unprecedented scope than ever, which significantly contributes to the success of deep learning. As each model family consists of pretrained models with diverse scales (e.g., DeiT-Ti/S/B), it naturally arises a fundamental question of how to efficiently assemble these readily available models in a family for dynamic accuracy-efficiency trade-offs at runtime. To this end, we present Stitchable Neural Networks (SN-Net), a novel scalable and efficient framework for model deployment. It cheaply produces numerous networks with different complexity and performance trade-offs given a family of pretrained neural networks, which we call anchors. Specifically, SN-Net splits the anchors across the blocks/layers and then stitches them together with simple stitching layers to map the activations from one anchor to another. With only a few epochs of training, SN-Net effectively interpolates between the performance of anchors with varying scales. At runtime, SN-Net can instantly adapt to dynamic resource constraints by switching the stitching positions. Extensive experiments on ImageNet classification demonstrate that SN-Net can obtain on-par or even better performance than many individually trained networks while supporting diverse deployment scenarios. For example, by stitching Swin Transformers, we challenge hundreds of models in Timm model zoo with a single network. We believe this new elastic model framework can serve as a strong baseline for further research in wider communities.

Recent advancements in building domain-specific large language models (LLMs) have shown remarkable success, especially in tasks requiring reasoning abilities like logical inference over complex relationships and multi-step problem solving. However, creating a powerful all-in-one LLM remains challenging due to the need for proprietary data and vast computational resources. As a resource-friendly alternative, we explore the potential of merging multiple expert models into a single LLM. Existing studies on model merging mainly focus on generalist LLMs instead of domain experts, or the LLMs under the same architecture and size. In this work, we propose an unconstrained model merging framework that accommodates both homogeneous and heterogeneous model architectures with a focus on reasoning tasks. A fine-grained layer-wise weight merging strategy is designed for homogeneous models merging, while heterogeneous model merging is built upon the probabilistic distribution knowledge derived from instruction-response fine-tuning data. Across 7 benchmarks and 9 reasoning-optimized LLMs, we reveal key findings that combinatorial reasoning emerges from merging which surpasses simple additive effects. We propose that unconstrained model merging could serve as a foundation for decentralized LLMs, marking a notable progression from the existing centralized LLM framework. This evolution could enhance wider participation and stimulate additional advancement in the field of artificial intelligence, effectively addressing the constraints posed by centralized models.

Merging finetuned Large Language Models (LLMs) has become increasingly important for integrating diverse capabilities into a single unified model. However, prevailing model merging methods rely on linear arithmetic in Euclidean space, which often destroys the intrinsic geometric properties of pretrained weights, such as hyperspherical energy. To address this, we propose Orthogonal Model Merging (OrthoMerge), a method that performs merging operations on the Riemannian manifold formed by the orthogonal group to preserve the geometric structure of the model's weights. By mapping task-specific orthogonal matrices learned by Orthogonal Finetuning (OFT) to the Lie algebra, OrthoMerge enables a principled yet efficient integration that takes into account both the direction and intensity of adaptations. In addition to directly leveraging orthogonal matrices obtained by OFT, we further extend this approach to general models finetuned with non-OFT methods (i.e., low-rank finetuning, full finetuning) via an Orthogonal-Residual Decoupling strategy. This technique extracts the orthogonal components of expert models by solving the orthogonal Procrustes problem, which are then merged on the manifold of the orthogonal group, while the remaining linear residuals are processed through standard additive merging. Extensive empirical results demonstrate the effectiveness of OrthoMerge in mitigating catastrophic forgetting and maintaining model performance across diverse tasks.

We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics poczos13aistats, to anomaly detection in piezometer data of embankment dams Jung15Exploration, to cosmology Ntampaka16Dynamical,Ravanbakhsh16ICML1. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

Model merging refers to the process of integrating multiple distinct models into a unified model that preserves and combines the strengths and capabilities of the individual models. Most existing approaches rely on task vectors to combine models, typically under the assumption that model parameters are accessible. However, for extremely large language models (LLMs) such as GPT-4, which are often provided solely as black-box services through API interfaces (Language-Model-as-a-Service), model weights are not available to end users. This presents a significant challenge, which we refer to as black-box model merging (BMM) with massive LLMs. To address this challenge, we propose a derivative-free optimization framework based on the evolutionary algorithm (Evo-Merging) that enables effective model merging using only inference-time API queries. Our method consists of two key components: (1) sparsity-based denoising, designed to identify and filter out irrelevant or redundant information across models, and (2) sign-aware scaling, which dynamically computes optimal combination weights for the relevant models based on their performance. We also provide a formal justification, along with a theoretical analysis, for our asymmetric sparsification. Extensive experimental evaluations demonstrate that our approach achieves state-of-the-art results on a range of tasks, significantly outperforming existing strong baselines.

When using Large Language Models (LLMs) to support Knowledge Graph Engineering (KGE), one of the first indications when searching for an appropriate model is its size. According to the scaling laws, larger models typically show higher capabilities. However, in practice, resource costs are also an important factor and thus it makes sense to consider the ratio between model performance and costs. The LLM-KG-Bench framework enables the comparison of LLMs in the context of KGE tasks and assesses their capabilities of understanding and producing KGs and KG queries. Based on a dataset created in an LLM-KG-Bench run covering 26 open state-of-the-art LLMs, we explore the model size scaling laws specific to KGE tasks. In our analyses, we assess how benchmark scores evolve between different model size categories. Additionally, we inspect how the general score development of single models and families of models correlates to their size. Our analyses revealed that, with a few exceptions, the model size scaling laws generally also apply to the selected KGE tasks. However, in some cases, plateau or ceiling effects occurred, i.e., the task performance did not change much between a model and the next larger model. In these cases, smaller models could be considered to achieve high cost-effectiveness. Regarding models of the same family, sometimes larger models performed worse than smaller models of the same family. These effects occurred only locally. Hence it is advisable to additionally test the next smallest and largest model of the same family.

Training a family of large language models targeting multiple scales and deployment objectives is prohibitively expensive, requiring separate training runs for each different size. Recent work on model compression through pruning and knowledge distillation has reduced this cost; however, this process still incurs hundreds of billions of tokens worth of training cost per compressed model. In this paper, we present Nemotron Elastic, a framework for building reasoning-oriented LLMs, including hybrid Mamba-Attention architectures, that embed multiple nested submodels within a single parent model, each optimized for different deployment configurations and budgets. Each of these submodels shares weights with the parent model and can be extracted zero-shot during deployment without additional training or fine-tuning. We enable this functionality through an end-to-end trained router, tightly coupled to a two-stage training curriculum designed specifically for reasoning models. We additionally introduce group-aware SSM elastification that preserves Mamba's structural constraints, heterogeneous MLP elastification, normalized MSE-based layer importance for improved depth selection, and knowledge distillation enabling simultaneous multi-budget optimization. We apply Nemotron Elastic to the Nemotron Nano V2 12B model, simultaneously producing a 9B and a 6B model using only 110B training tokens; this results in over 360x cost reduction compared to training model families from scratch, and around 7x compared to SoTA compression techniques. Each of the nested models performs on par or better than the SoTA in accuracy. Moreover, unlike other compression methods, the nested capability of our approach allows having a many-in-one reasoning model that has constant deployment memory against the number of models in the family.

Model merging is an efficient empowerment technique in the machine learning community that does not require the collection of raw training data and does not require expensive computation. As model merging becomes increasingly prevalent across various fields, it is crucial to understand the available model merging techniques comprehensively. However, there is a significant gap in the literature regarding a systematic and thorough review of these techniques. This survey provides a comprehensive overview of model merging methods and theories, their applications in various domains and settings, and future research directions. Specifically, we first propose a new taxonomic approach that exhaustively discusses existing model merging methods. Secondly, we discuss the application of model merging techniques in large language models, multimodal large language models, and 10+ machine learning subfields, including continual learning, multi-task learning, few-shot learning, etc. Finally, we highlight the remaining challenges of model merging and discuss future research directions. A comprehensive list of papers about model merging is available at https://github.com/EnnengYang/Awesome-Model-Merging-Methods-Theories-Applications.

Large Language Models (LLMs) face significant challenges regarding deployment costs and latency, necessitating adaptive computing strategies. Building upon the AI Flow framework, we introduce Ruyi2 as an evolution of our adaptive model series designed for efficient variable-depth computation. While early-exit architectures offer a viable efficiency-performance balance, the Ruyi model and existing methods often struggle with optimization complexity and compatibility with large-scale distributed training. To bridge this gap, Ruyi2 introduces a stable "Familial Model" based on Megatron-LM. By using 3D parallel training, it achieves a 2-3 times speedup over Ruyi, while performing comparably to same-sized Qwen3 models. These results confirm that family-based parameter sharing is a highly effective strategy, establishing a new "Train Once, Deploy Many" paradigm and providing a key reference for balancing architectural efficiency with high-performance capabilities.

Model merging aims to combine multiple fine-tuned models into a single set of weights that performs well across all source tasks. While prior work has shown that merging can approximate the performance of individual fine-tuned models for each task, it largely overlooks scenarios where models are compressed into low-rank representations, either through low-rank adaptation (LoRA) or post-training singular value decomposition (SVD). We first demonstrate that applying conventional merging methods to low-rank weights leads to severe performance degradation in the merged model. Motivated by this phenomenon, we propose a fundamentally different approach: instead of collapsing all adapters into one set of weights, we construct a compact basis (e.g., an equivalent of holding two or more models) from which original task-specific models can be recovered via linear combination. This reframes merging as generating a reconstruction-capable model space rather than producing a single merged model. Crucially, this allows us to ``revert'' to each individual model when needed, recognizing that no merged model can consistently outperform one specialized for its task. Building on this insight, we introduce our method, Reversible Model Merging (RMM), an efficient, data-free, and flexible method that provides a closed-form solution for selecting the optimal basis of model weights and task-specific coefficients for linear combination. Extensive experiments across diverse datasets and model scales demonstrate that RMM consistently outperforms existing merging approaches, preserving the performance of low-rank compressed models by a significant margin.

In this paper, we propose a novel learning paradigm, termed Chain-of-Model (CoM), which incorporates the causal relationship into the hidden states of each layer as a chain style, thereby introducing great scaling efficiency in model training and inference flexibility in deployment. We introduce the concept of Chain-of-Representation (CoR), which formulates the hidden states at each layer as a combination of multiple sub-representations (i.e., chains) at the hidden dimension level. In each layer, each chain from the output representations can only view all of its preceding chains in the input representations. Consequently, the model built upon CoM framework can progressively scale up the model size by increasing the chains based on the previous models (i.e., chains), and offer multiple sub-models at varying sizes for elastic inference by using different chain numbers. Based on this principle, we devise Chain-of-Language-Model (CoLM), which incorporates the idea of CoM into each layer of Transformer architecture. Based on CoLM, we further introduce CoLM-Air by introducing a KV sharing mechanism, that computes all keys and values within the first chain and then shares across all chains. This design demonstrates additional extensibility, such as enabling seamless LM switching, prefilling acceleration and so on. Experimental results demonstrate our CoLM family can achieve comparable performance to the standard Transformer, while simultaneously enabling greater flexiblity, such as progressive scaling to improve training efficiency and offer multiple varying model sizes for elastic inference, paving a a new way toward building language models. Our code will be released in the future at: https://github.com/microsoft/CoLM.

Model merging provides a scalable alternative to multi-task training by combining specialized finetuned models through parameter arithmetic, enabling efficient deployment without the need for joint training or access to all task data. While recent methods have shown promise, existing evaluations are limited in both model scale and task diversity, leaving open questions about their applicability to large, domain-specialized LLMs. To tackle the challenges, we introduce MergeBench, a comprehensive evaluation suite designed to assess model merging at scale. MergeBench builds on state-of-the-art open-source language models, including Llama and Gemma families at 2B to 9B scales, and covers five key domains: instruction following, mathematics, multilingual understanding, coding and safety. We standardize finetuning and evaluation protocols, and assess eight representative merging methods across multi-task performance, forgetting and runtime efficiency. Based on extensive experiments, we provide practical guidelines for algorithm selection and share insights showing that model merging tends to perform better on stronger base models, with techniques such as merging coefficient tuning and sparsification improving knowledge retention. However, several challenges remain, including the computational cost on large models, the gap for in-domain performance compared to multi-task models, and the underexplored role of model merging in standard LLM training pipelines. We hope MergeBench provides a foundation for future research to advance the understanding and practical application of model merging. Our project page is at https://yifei-he.github.io/mergebench/{https://yifei-he.github.io/mergebench/}.

In recent years, Graph Neural Networks (GNNs) have made significant advances in processing structured data. However, most of them primarily adopted a model-centric approach, which simplifies graphs by converting them into undirected formats and emphasizes model designs. This approach is inherently limited in real-world applications due to the unavoidable information loss in simple undirected graphs and the model optimization challenges that arise when exceeding the upper bounds of this sub-optimal data representational capacity. As a result, there has been a shift toward data-centric methods that prioritize improving graph quality and representation. Specifically, various types of graphs can be derived from naturally structured data, including heterogeneous graphs, hypergraphs, and directed graphs. Among these, directed graphs offer distinct advantages in topological systems by modeling causal relationships, and directed GNNs have been extensively studied in recent years. However, a comprehensive survey of this emerging topic is still lacking. Therefore, we aim to provide a comprehensive review of directed graph learning, with a particular focus on a data-centric perspective. Specifically, we first introduce a novel taxonomy for existing studies. Subsequently, we re-examine these methods from the data-centric perspective, with an emphasis on understanding and improving data representation. It demonstrates that a deep understanding of directed graphs and their quality plays a crucial role in model performance. Additionally, we explore the diverse applications of directed GNNs across 10+ domains, highlighting their broad applicability. Finally, we identify key opportunities and challenges within the field, offering insights that can guide future research and development in directed graph learning.

Neural networks can be significantly compressed by pruning, yielding sparse models with reduced storage and computational demands while preserving predictive performance. Model soups (Wortsman et al., 2022) enhance generalization and out-of-distribution (OOD) performance by averaging the parameters of multiple models into a single one, without increasing inference time. However, achieving both sparsity and parameter averaging is challenging as averaging arbitrary sparse models reduces the overall sparsity due to differing sparse connectivities. This work addresses these challenges by demonstrating that exploring a single retraining phase of Iterative Magnitude Pruning (IMP) with varied hyperparameter configurations such as batch ordering or weight decay yields models suitable for averaging, sharing identical sparse connectivity by design. Averaging these models significantly enhances generalization and OOD performance over their individual counterparts. Building on this, we introduce Sparse Model Soups (SMS), a novel method for merging sparse models by initiating each prune-retrain cycle with the averaged model from the previous phase. SMS preserves sparsity, exploits sparse network benefits, is modular and fully parallelizable, and substantially improves IMP's performance. We further demonstrate that SMS can be adapted to enhance state-of-the-art pruning-during-training approaches.

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families {S_{i}}_{i in I} of contractions on a Hilbert space H, to commuting families {T_{i}}_{i in I} of contractive C_{0}-semigroups on L^{2}(prod_{i in I}T) otimes H. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for d in N with d geq 3 the existence of commuting families {T_{i}}_{i=1}^{d} of contractive C_{0}-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of R_{geq 0}^{d} of non-unitarily dilatable and non-unitarily approximable d-parameter contractive C_{0}-semigroups on separable infinite-dimensional Hilbert spaces for each d geq 3. Similar results are also developed for d-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, viz. that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of C_{0}-semigroups, which extends results of Eisner (2009--10).

We formalize constraint-based structure learning of the "true" causal graph from observed data when unobserved variables are also existent. We provide conditions for a "natural" family of constraint-based structure-learning algorithms that output graphs that are Markov equivalent to the causal graph. Under the faithfulness assumption, this natural family contains all exact structure-learning algorithms. We also provide a set of assumptions, under which any natural structure-learning algorithm outputs Markov equivalent graphs to the causal graph. These assumptions can be thought of as a relaxation of faithfulness, and most of them can be directly tested from (the underlying distribution) of the data, particularly when one focuses on structural causal models. We specialize the definitions and results for structural causal models.

Model merging is a powerful technique for integrating the specialized knowledge of multiple machine learning models into a single model. However, existing methods require manually partitioning model parameters into fixed groups for merging, which restricts the exploration of potential combinations and limits performance. To overcome these limitations, we propose Model Merging of Natural Niches (M2N2), an evolutionary algorithm with three key features: (1) dynamic adjustment of merging boundaries to progressively explore a broader range of parameter combinations; (2) a diversity preservation mechanism inspired by the competition for resources in nature, to maintain a population of diverse, high-performing models that are particularly well-suited for merging; and (3) a heuristicbased attraction metric to identify the most promising pairs of models for fusion. Our experimental results demonstrate, for the first time, that model merging can be used to evolve models entirely from scratch. Specifically, we apply M2N2 to evolve MNIST classifiers from scratch and achieve performance comparable to CMA-ES, while being computationally more efficient. Furthermore, M2N2 scales to merge specialized language and image generation models, achieving state-of-the-art performance. Notably, it preserves crucial model capabilities beyond those explicitly optimized by the fitness function, highlighting its robustness and versatility. Our code is available at https://github.com/SakanaAI/natural_niches

We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically involves an underlying projection process that maps the original hypergraph onto a graph, and is common in graph-based analysis. While hypergraph projection can potentially lead to loss of higher-order relations, there exists very limited studies on the consequences of doing so, as well as its remediation. This work fills this gap by doing two things: (1) we develop analysis based on graph and set theory, showing two ubiquitous patterns of hyperedges that are root to structural information loss in all hypergraph projections; we also quantify the combinatorial impossibility of recovering the lost higher-order structures if no extra help is provided; (2) we still seek to recover the lost higher-order structures in hypergraph projection, and in light of (1)'s findings we propose to relax the problem into a learning-based setting. Under this setting, we develop a learning-based hypergraph reconstruction method based on an important statistic of hyperedge distributions that we find. Our reconstruction method is evaluated on 8 real-world datasets under different settings, and exhibits consistently good performance. We also demonstrate benefits of the reconstructed hypergraphs via use cases of protein rankings and link predictions.

We present Ruyi2.5, a multimodal familial model built on the AI Flow framework. Extending Ruyi2's "Train Once, Deploy Many" paradigm to the multimodal domain, Ruyi2.5 constructs a shared-backbone architecture that co-trains models of varying scales within a single unified pipeline, ensuring semantic consistency across all deployment tiers. Built upon Ruyi2.5, Ruyi2.5-Camera model is developed as a privacy-preserving camera service system, which instantiates Ruyi2.5-Camera into a two-stage recognition pipeline: an edge model applies information-bottleneck-guided irreversible feature mapping to de-identify raw frames at the source, while a cloud model performs deep behavior reasoning. To accelerate reinforcement learning fine-tuning, we further propose Binary Prefix Policy Optimization (BPPO), which reduces sample redundancy via binary response selection and focuses gradient updates on response prefixes, achieving a 2 to 3 times training speedup over GRPO. Experiments show Ruyi2.5 matches Qwen3-VL on the general multimodal benchmarks, while Ruyi2.5-Camera substantially outperforms Qwen3-VL on privacy-constrained surveillance tasks.

Recent work in probability-domain knowledge distillation has established axiomatic frameworks for temperature scaling, multi-teacher aggregation, and bias-variance trade-offs in single-stage settings. However, the mathematical behavior of recursive or multi-generation distillation remains poorly understood, with prior approaches relying primarily on empirical heuristics. In this work, we introduce an axiomatic and operator-theoretic framework for recursive meta-distillation, formalizing iterative knowledge distillation as a sequence of probability-distribution operators with explicit anchoring to base teachers. We define structural axioms for valid meta-teacher construction and prove the existence of non-trivial operator families satisfying these axioms without specifying particular algorithms or loss functions. Under mild realizability and convexity assumptions, we show that anchored recursive distillation induces contraction in KL divergence, yielding geometric convergence to base teacher distributions and a unique, globally attractive fixed point. The contribution is foundational rather than algorithmic: the framework characterizes when recursive distillation is mathematically well-posed and convergent rather than error-accumulating, independent of model architecture, optimization details, or specific operator instantiations. These results provide a theoretical basis for understanding stability, bias-variance behavior, and failure modes in iterative and multi-teacher distillation under capacity constraints.

Model merging has emerged as a promising approach for multi-task learning (MTL), offering a data-efficient alternative to conventional fine-tuning. However, with the rapid development of the open-source AI ecosystem and the increasing availability of fine-tuned foundation models, existing model merging methods face two key limitations: (i) They are primarily designed for in-house fine-tuned models, making them less adaptable to diverse model sources with partially unknown model and task information, (ii) They struggle to scale effectively when merging numerous model checkpoints. To address these challenges, we formulate model merging as a constrained optimization problem and introduce a novel approach: Frank-Wolfe Merging (FW-Merging). Inspired by Frank-Wolfe optimization, our approach iteratively selects the most relevant model in the pool to minimize a linear approximation of the objective function and then executes a local merging similar to the Frank-Wolfe update. The objective function is designed to capture the desired behavior of the target-merged model, while the fine-tuned candidate models define the constraint set. More importantly, FW-Merging serves as an orthogonal technique for existing merging methods, seamlessly integrating with them to further enhance accuracy performance. Our experiments show that FW-Merging scales across diverse model sources, remaining stable with 16 irrelevant models and improving by 15.3% with 16 relevant models on 20 CV tasks, while maintaining constant memory overhead, unlike the linear overhead of data-informed merging methods. Compared with the state-of-the-art approaches, FW-Merging surpasses the data-free merging method by 32.8% and outperforms the data-informed Adamerging by 8.39% when merging 20 ViT models. Our code is open-sourced at github.com/hmarkc/FW-Merging.

Recently, Ainsworth et al. showed that using weight matching (WM) to minimize the L_2 distance in a permutation search of model parameters effectively identifies permutations that satisfy linear mode connectivity (LMC), in which the loss along a linear path between two independently trained models with different seeds remains nearly constant. This paper provides a theoretical analysis of LMC using WM, which is crucial for understanding stochastic gradient descent's effectiveness and its application in areas like model merging. We first experimentally and theoretically show that permutations found by WM do not significantly reduce the L_2 distance between two models and the occurrence of LMC is not merely due to distance reduction by WM in itself. We then provide theoretical insights showing that permutations can change the directions of the singular vectors, but not the singular values, of the weight matrices in each layer. This finding shows that permutations found by WM mainly align the directions of singular vectors associated with large singular values across models. This alignment brings the singular vectors with large singular values, which determine the model functionality, closer between pre-merged and post-merged models, so that the post-merged model retains functionality similar to the pre-merged models, making it easy to satisfy LMC. Finally, we analyze the difference between WM and straight-through estimator (STE), a dataset-dependent permutation search method, and show that WM outperforms STE, especially when merging three or more models.

Requiring 10^{13}-10^{15} FLOPs to calculate one 8 bit weight in an LLM during pretraining is extremely expensive and seems inefficient. To better leverage the huge investments made into pretrained models, we develop the new "Assembly-of-Experts" (AoE) construction method to create capable child variants of existing Mixture-of-Experts parent models in linear time. Model weight tensors get interpolated individually, allowing to enhance or suppress semantic features of the parents. Varying the proportion of weights taken from the parent models, we observe some properties of the AoE child model changing gradually, while other behavioral traits emerge with a sharp transition. Surprisingly, nearly every generated model is functional and capable, which makes searching the model space straightforward. We construct the DeepSeek R1T "Chimera", a 671B open-weights hybrid model combining DeepSeek's V3-0324 and R1 model variants. The child inherits only the routed expert tensors of R1, but still achieves about R1-level intelligence. At the same time, it uses about 40\% fewer output tokens, close to V3 speed. Constructed without any fine-tuning or distillation, the Chimera exhibits surprisingly compact, orderly reasoning compared to its parent models.

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph G=(V, D,B) is parameterized by a rational function with parameters for each vertex and edge in G. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when D, the directed part of the mixed graph G, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from D through some small operations. This closed form expression then allows us to show that if G is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

Scaling laws for large language models (LLMs) predict model performance based on parameters like size and training data. However, differences in training configurations and data processing across model families lead to significant variations in benchmark performance, making it difficult for a single scaling law to generalize across all LLMs. On the other hand, training family-specific scaling laws requires training models of varying sizes for every family. In this work, we propose Skills Scaling Laws (SSLaws, pronounced as Sloth), a novel scaling law that leverages publicly available benchmark data and assumes LLM performance is driven by low-dimensional latent skills, such as reasoning and instruction following. These latent skills are influenced by computational resources like model size and training tokens but with varying efficiencies across model families. Sloth exploits correlations across benchmarks to provide more accurate and interpretable predictions while alleviating the need to train multiple LLMs per family. We present both theoretical results on parameter identification and empirical evaluations on 12 prominent benchmarks, from Open LLM Leaderboard v1/v2, demonstrating that Sloth predicts LLM performance efficiently and offers insights into scaling behaviors for complex downstream tasks and increased test-time compute.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

Model merging has emerged as an effective approach to combine multiple single-task models into a multitask model. This process typically involves computing a weighted average of the model parameters without any additional training. Existing model-merging methods focus on enhancing average task accuracy. However, interference and conflicts between the objectives of different tasks can lead to trade-offs during the merging process. In real-world applications, a set of solutions with various trade-offs can be more informative, helping practitioners make decisions based on diverse preferences. In this paper, we introduce a novel and low-compute algorithm, Model Merging with Amortized Pareto Front (MAP). MAP efficiently identifies a Pareto set of scaling coefficients for merging multiple models, reflecting the trade-offs involved. It amortizes the substantial computational cost of evaluations needed to estimate the Pareto front by using quadratic approximation surrogate models derived from a pre-selected set of scaling coefficients. Experimental results on vision and natural language processing tasks demonstrate that MAP can accurately identify the Pareto front, providing practitioners with flexible solutions to balance competing task objectives. We also introduce Bayesian MAP for scenarios with a relatively low number of tasks and Nested MAP for situations with a high number of tasks, further reducing the computational cost of evaluation.

For a positive integer k, a graph property H, and a graph parameter P, let ex_{P}(n, H; δgeq k) denote the maximum value of P over all n-vertex graphs with minimum degree at least k that do not possess the property H. The corresponding extremal families are denoted by EX_{P}(n, H; δgeq k). For two disjoint graphs H_1 and H_2, let H_1 cup H_2 denote their (disjoint) union, i.e., the graph with vertex set V(H_1) cup V(H_2) and edge set E(H_1) cup E(H_2); and let H_1 vee H_2 denote their join. In 1962, Erdős established a classical theorem on the maximum number of edges in a non-Hamiltonian graph of given order and minimum degree. Motivated by recent work on feasible graph parameters in Ai2023, we prove several extensions of Erdős's 1962 theorem on non-Hamiltonian graphs. The first result gives a common generalization of the extremal theorem due to Erdős and its spectral analogs. As direct applications, we obtain complete solutions to open problems raised in the literature since 2016, thereby improving nearly all related prior results in this direction. Our proof technique differs somewhat from those in MR3539577,MR3556876. We also prove an analog theorem for the Hamiltonian-connected property and obtain a result which extends the theorem of Füredi, Kostochka, and Luo MR3843180 on Hamilton cycles.

The emergence of large-scale multi-modal generative models has drastically advanced artificial intelligence, introducing unprecedented levels of performance and functionality. However, optimizing these models remains challenging due to historically isolated paths of model-centric and data-centric developments, leading to suboptimal outcomes and inefficient resource utilization. In response, we present a novel sandbox suite tailored for integrated data-model co-development. This sandbox provides a comprehensive experimental platform, enabling rapid iteration and insight-driven refinement of both data and models. Our proposed "Probe-Analyze-Refine" workflow, validated through applications on state-of-the-art LLaVA-like and DiT based models, yields significant performance boosts, such as topping the VBench leaderboard. We also uncover fruitful insights gleaned from exhaustive benchmarks, shedding light on the critical interplay between data quality, diversity, and model behavior. With the hope of fostering deeper understanding and future progress in multi-modal data and generative modeling, our codes, datasets, and models are maintained and accessible at https://github.com/modelscope/data-juicer/blob/main/docs/Sandbox.md.

Generative models trained on internet-scale data are capable of generating novel and realistic texts, images, and videos. A natural next question is whether these models can advance science, for example by generating novel stable materials. Traditionally, models with explicit structures (e.g., graphs) have been used in modeling structural relationships in scientific data (e.g., atoms and bonds in crystals), but generating structures can be difficult to scale to large and complex systems. Another challenge in generating materials is the mismatch between standard generative modeling metrics and downstream applications. For instance, common metrics such as the reconstruction error do not correlate well with the downstream goal of discovering stable materials. In this work, we tackle the scalability challenge by developing a unified crystal representation that can represent any crystal structure (UniMat), followed by training a diffusion probabilistic model on these UniMat representations. Our empirical results suggest that despite the lack of explicit structure modeling, UniMat can generate high fidelity crystal structures from larger and more complex chemical systems, outperforming previous graph-based approaches under various generative modeling metrics. To better connect the generation quality of materials to downstream applications, such as discovering novel stable materials, we propose additional metrics for evaluating generative models of materials, including per-composition formation energy and stability with respect to convex hulls through decomposition energy from Density Function Theory (DFT). Lastly, we show that conditional generation with UniMat can scale to previously established crystal datasets with up to millions of crystals structures, outperforming random structure search (the current leading method for structure discovery) in discovering new stable materials.

Typical deep visual recognition models are capable of performing the one task they were trained on. In this paper, we tackle the extremely difficult problem of combining completely distinct models with different initializations, each solving a separate task, into one multi-task model without any additional training. Prior work in model merging permutes one model to the space of the other then adds them together. While this works for models trained on the same task, we find that this fails to account for the differences in models trained on disjoint tasks. Thus, we introduce "ZipIt!", a general method for merging two arbitrary models of the same architecture that incorporates two simple strategies. First, in order to account for features that aren't shared between models, we expand the model merging problem to additionally allow for merging features within each model by defining a general "zip" operation. Second, we add support for partially zipping the models up until a specified layer, naturally creating a multi-head model. We find that these two changes combined account for a staggering 20-60% improvement over prior work, making the merging of models trained on disjoint tasks feasible.

Weight-space merging aims to combine multiple fine-tuned LLMs into a single model without retraining, yet most existing approaches remain fundamentally parameter-space heuristics. This creates three practical limitations. First, linear averaging, task vectors, and related rules operate on Euclidean coordinates, even though the desired goal is to merge functionality, i.e., predictive behaviors across tasks. Second, when the source checkpoints are farther apart or more heterogeneous, Euclidean blends often trigger representation collapse, manifested as activation variance shrinkage and effective-rank degradation, which sharply degrades accuracy. Third, many geometry-inspired methods are most natural for two-model interpolation and do not extend cleanly to merging N>2 experts with a principled objective. We address these issues by formulating model merging as computing a weighted Karcher mean on the Fisher--Rao manifold, which is locally equivalent to minimizing a KL-based function distance between predictive distributions. We derive a practical fixed-point algorithm using a lightweight spherical proxy that preserves norms and generalizes directly to multi-expert merging. Across various benchmarks and collapse diagnostics, our method remains stable as the number and heterogeneity of merged models increase, consistently outperforming prior baselines.

Model merging allows combining the capabilities of existing models into a new one - post hoc, without additional training. This has made it increasingly popular thanks to its low cost and the availability of libraries that support merging on consumer GPUs. Recent work shows that pairing merging with evolutionary algorithms can boost performance, but no framework currently supports flexible experimentation with such strategies in language models. We introduce Mergenetic, an open-source library for evolutionary model merging. Mergenetic enables easy composition of merging methods and evolutionary algorithms while incorporating lightweight fitness estimators to reduce evaluation costs. We describe its design and demonstrate that Mergenetic produces competitive results across tasks and languages using modest hardware.

Model merging has emerged as an efficient and flexible paradigm for multi-task learning, with numerous methods being proposed in recent years. However, these state-of-the-art techniques are typically evaluated on benchmark suites that are highly favorable to model merging, and their robustness in more realistic settings remains largely unexplored. In this work, we first investigate the vulnerabilities of model-merging methods and pinpoint the source-model characteristics that critically underlie them. Specifically, we identify two factors that are particularly harmful to the merging process: (1) disparities in task vector norms, and (2) the low confidence of the source models. To address this issue, we propose DisTaC (Distillation for Task vector Conditioning), a novel method that pre-conditions these problematic task vectors before the merge. DisTaC leverages knowledge distillation to adjust a task vector's norm and increase source-model confidence while preserving its essential task-specific knowledge. Our extensive experiments demonstrate that by pre-conditioning task vectors with DisTaC, state-of-the-art merging techniques can successfully integrate models exhibiting the harmful traits -- where they would otherwise fail -- achieving significant performance gains.

In the domain of model-based engineering, models are essential components that enable system design and analysis. Traditionally, the creation of these models has been a manual process requiring not only deep modeling expertise but also substantial domain knowledge of target systems. With the rapid advancement of generative artificial intelligence, large language models (LLMs) show potential for automating model generation. This work explores the generation of instance models using LLMs, focusing specifically on producing XMI-based instance models from Ecore metamodels and natural language specifications. We observe that current LLMs struggle to directly generate valid XMI models. To address this, we propose a two-step approach: first, using LLMs to produce a simplified structured output containing all necessary instance model information, namely a conceptual instance model, and then compiling this intermediate representation into a valid XMI file. The conceptual instance model is format-independent, allowing it to be transformed into various modeling formats via different compilers. The feasibility of the proposed method has been demonstrated using several LLMs, including GPT-4o, o1-preview, Llama 3.1 (8B and 70B). Results show that the proposed method significantly improves the usability of LLMs for instance model generation tasks. Notably, the smaller open-source model, Llama 3.1 70B, demonstrated performance comparable to proprietary GPT models within the proposed framework.

Linear Mode Connectivity (LMC) refers to the phenomenon that performance remains consistent for linearly interpolated models in the parameter space. For independently optimized model pairs from different random initializations, achieving LMC is considered crucial for validating the stable success of the non-convex optimization in modern machine learning models and for facilitating practical parameter-based operations such as model merging. While LMC has been achieved for neural networks by considering the permutation invariance of neurons in each hidden layer, its attainment for other models remains an open question. In this paper, we first achieve LMC for soft tree ensembles, which are tree-based differentiable models extensively used in practice. We show the necessity of incorporating two invariances: subtree flip invariance and splitting order invariance, which do not exist in neural networks but are inherent to tree architectures, in addition to permutation invariance of trees. Moreover, we demonstrate that it is even possible to exclude such additional invariances while keeping LMC by designing decision list-based tree architectures, where such invariances do not exist by definition. Our findings indicate the significance of accounting for architecture-specific invariances in achieving LMC.

Robot design aims at learning to create robots that can be easily controlled and perform tasks efficiently. Previous works on robot design have proven its ability to generate robots for various tasks. However, these works searched the robots directly from the vast design space and ignored common structures, resulting in abnormal robots and poor performance. To tackle this problem, we propose a Symmetry-Aware Robot Design (SARD) framework that exploits the structure of the design space by incorporating symmetry searching into the robot design process. Specifically, we represent symmetries with the subgroups of the dihedral group and search for the optimal symmetry in structured subgroups. Then robots are designed under the searched symmetry. In this way, SARD can design efficient symmetric robots while covering the original design space, which is theoretically analyzed. We further empirically evaluate SARD on various tasks, and the results show its superior efficiency and generalizability.

We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, NTP_2, NSOP_1, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between NSOP_1 and Kruckman and Ramsey's BTP that applies to most known NSOP_4 examples in the literature.

We present a novel application of evolutionary algorithms to automate the creation of powerful foundation models. While model merging has emerged as a promising approach for LLM development due to its cost-effectiveness, it currently relies on human intuition and domain knowledge, limiting its potential. Here, we propose an evolutionary approach that overcomes this limitation by automatically discovering effective combinations of diverse open-source models, harnessing their collective intelligence without requiring extensive additional training data or compute. Our approach operates in both parameter space and data flow space, allowing for optimization beyond just the weights of the individual models. This approach even facilitates cross-domain merging, generating models like a Japanese LLM with Math reasoning capabilities. Surprisingly, our Japanese Math LLM achieved state-of-the-art performance on a variety of established Japanese LLM benchmarks, even surpassing models with significantly more parameters, despite not being explicitly trained for such tasks. Furthermore, a culturally-aware Japanese VLM generated through our approach demonstrates its effectiveness in describing Japanese culture-specific content, outperforming previous Japanese VLMs. This work not only contributes new state-of-the-art models back to the open-source community, but also introduces a new paradigm for automated model composition, paving the way for exploring alternative, efficient approaches to foundation model development.

Model merging integrates the weights of multiple task-specific models into a single multi-task model. Despite recent interest in the problem, a significant performance gap between the combined and single-task models remains. In this paper, we investigate the key characteristics of task matrices -- weight update matrices applied to a pre-trained model -- that enable effective merging. We show that alignment between singular components of task-specific and merged matrices strongly correlates with performance improvement over the pre-trained model. Based on this, we propose an isotropic merging framework that flattens the singular value spectrum of task matrices, enhances alignment, and reduces the performance gap. Additionally, we incorporate both common and task-specific subspaces to further improve alignment and performance. Our proposed approach achieves state-of-the-art performance across multiple scenarios, including various sets of tasks and model scales. This work advances the understanding of model merging dynamics, offering an effective methodology to merge models without requiring additional training. Code is available at https://github.com/danielm1405/iso-merging .

This paper makes a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which results in the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. A second important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

Model merging aims to integrate multiple task-adapted models into a unified model that preserves the knowledge of each task. In this paper, we identify that the key to this knowledge retention lies in maintaining the directional consistency of singular spaces between merged multi-task vector and individual task vectors. However, this consistency is frequently compromised by two issues: i) an imbalanced energy distribution within task vectors, where a small fraction of singular values dominate the total energy, leading to the neglect of semantically important but weaker components upon merging, and ii) the geometric inconsistency of task vectors in parameter space, which causes direct merging to distort their underlying directional geometry. To address these challenges, we propose DC-Merge, a method for directional-consistent model merging. It first balances the energy distribution of each task vector by smoothing its singular values, ensuring all knowledge components are adequately represented. These energy-balanced vectors are then projected onto a shared orthogonal subspace to align their directional geometries with minimal reconstruction error. Finally, the aligned vectors are aggregated in the shared orthogonal subspace and projected back to the original parameter space. Extensive experiments on vision and vision-language benchmarks show that DC-Merge consistently achieves state-of-the-art performance in both full fine-tuning and LoRA settings. The implementation code is available at https://github.com/Tobeginwith/DC-Merge.

By merging models, AI systems can combine the distinct strengths of separate language models, achieving a balance between multiple capabilities without requiring substantial retraining. However, the integration process can be intricate due to differences in training methods and fine-tuning, typically necessitating specialized knowledge and repeated refinement. This paper explores model merging techniques across a spectrum of complexity, examining where automated methods like evolutionary strategies stand compared to hyperparameter-driven approaches such as DARE, TIES-Merging and simpler methods like Model Soups. In addition, we introduce Differentiable Adaptive Merging (DAM), an efficient, adaptive merging approach as an alternative to evolutionary merging that optimizes model integration through scaling coefficients, minimizing computational demands. Our findings reveal that even simple averaging methods, like Model Soups, perform competitively when model similarity is high, underscoring each technique's unique strengths and limitations. We open-sourced DAM, including the implementation code and experiment pipeline, on GitHub: https://github.com/arcee-ai/DAM.

Knowledge graphs are incomplete by nature, with only a limited number of observed facts from the world knowledge being represented as structured relations between entities. To partly address this issue, an important task in statistical relational learning is that of link prediction or knowledge graph completion. Both linear and non-linear models have been proposed to solve the problem. Bilinear models, while expressive, are prone to overfitting and lead to quadratic growth of parameters in number of relations. Simpler models have become more standard, with certain constraints on bilinear map as relation parameters. In this work, we propose a factorized bilinear pooling model, commonly used in multi-modal learning, for better fusion of entities and relations, leading to an efficient and constraint-free model. We prove that our model is fully expressive, providing bounds on the embedding dimensionality and factorization rank. Our model naturally generalizes Tucker decomposition based TuckER model, which has been shown to generalize other models, as efficient low-rank approximation without substantially compromising the performance. Due to low-rank approximation, the model complexity can be controlled by the factorization rank, avoiding the possible cubic growth of TuckER. Empirically, we evaluate on real-world datasets, reaching on par or state-of-the-art performance. At extreme low-ranks, model preserves the performance while staying parameter efficient.

With the proliferation of domain-specific models, model merging has emerged as a set of techniques that combine the capabilities of multiple models into one that can multitask without the cost of additional training. In this paper, we propose a new model merging technique, Drop and rEscaLe via sampLing with mAgnitude (DELLA-Merging), that employs a novel pruning technique, MAGPRUNE, which shows significant advantages over DARE and TIES. MAGPRUNE first ranks the parameters in order of their magnitude and assigns higher dropout probabilities (p) to parameters with lower ranks corresponding to lower magnitudes. To approximate the original embeddings, MAGPRUNE employs a rescaling operation on the parameters that survive the random dropping by 1/(1 - p). On three different expert models considered for merging (LM, Math, Code) and corresponding benchmark datasets (AlpacaEval, GSM8K, MBPP), DELLA shows an average improvement of 2.4 points over baseline methods employing delta parameter pruning (an improvement of 3.6 points over TIES, 1.2 points over DARE), and 11.1 points over the no-pruning baseline (TA). We release the source code at: https://github.com/declare-lab/della.

Disentangling the factors of variation in data is a fundamental concept in machine learning and has been studied in various ways by different researchers, leading to a multitude of definitions. Despite the numerous empirical studies, more theoretical research is needed to fully understand the defining properties of disentanglement and how different definitions relate to each other. This paper presents a meta-analysis of existing definitions of disentanglement, using category theory as a unifying and rigorous framework. We propose that the concepts of the cartesian and monoidal products should serve as the core of disentanglement. With these core concepts, we show the similarities and crucial differences in dealing with (i) functions, (ii) equivariant maps, (iii) relations, and (iv) stochastic maps. Overall, our meta-analysis deepens our understanding of disentanglement and its various formulations and can help researchers navigate different definitions and choose the most appropriate one for their specific context.

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

The availability of performant pre-trained models has led to a proliferation of fine-tuned expert models that are specialized to a particular domain or task. Model MoErging methods aim to recycle expert models to create an aggregate system with improved performance or generalization. A key component of MoErging methods is the creation of a router that decides which expert model(s) to use for a particular input or application. The promise, effectiveness, and large design space of MoErging has spurred the development of many new methods over the past few years. This rapid pace of development has made it challenging to compare different MoErging methods, which are rarely compared to one another and are often validated in different experimental setups. To remedy such gaps, we present a comprehensive survey of MoErging methods that includes a novel taxonomy for cataloging key design choices and clarifying suitable applications for each method. Apart from surveying MoErging research, we inventory software tools and applications that make use of MoErging. We additionally discuss related fields of study such as model merging, multitask learning, and mixture-of-experts models. Taken as a whole, our survey provides a unified overview of existing MoErging methods and creates a solid foundation for future work in this burgeoning field.

We present FreeBird, an extensible Julia-based platform for computational studies of phase equilibria at generic interfaces. The package supports a range of system configurations, from atomistic solid surfaces to coarse-grained lattice-gas models, with energies evaluated using classical interatomic potentials or lattice Hamiltonians. Both atomistic and lattice systems accommodate single- or multi-component mixtures with flexibly definable surface and lattice geometries. Implemented sampling algorithms include nested sampling, Wang-Landau sampling, Metropolis Monte Carlo, and, for tractable lattice systems, exact enumeration. Leveraging Julia's type hierarchies and multiple dispatch, FreeBird provides a modular interface that allows seamless integration of system definitions, energy evaluators, and sampling schemes. Designed for flexibility, extensibility, and performance, FreeBird offers a versatile framework for exploring the thermodynamics of interfacial phenomena.

Pioneered by the foundational information theory by Claude Shannon and the visionary framework of machine intelligence by Alan Turing, the convergent evolution of information and communication technologies (IT/CT) has created an unbroken wave of connectivity and computation. This synergy has sparked a technological revolution, now reaching its peak with large artificial intelligence (AI) models that are reshaping industries and redefining human-machine collaboration. However, the realization of ubiquitous intelligence faces considerable challenges due to substantial resource consumption in large models and high communication bandwidth demands. To address these challenges, AI Flow has been introduced as a multidisciplinary framework that integrates cutting-edge IT and CT advancements, with a particular emphasis on the following three key points. First, device-edge-cloud framework serves as the foundation, which integrates end devices, edge servers, and cloud clusters to optimize scalability and efficiency for low-latency model inference. Second, we introduce the concept of familial models, which refers to a series of different-sized models with aligned hidden features, enabling effective collaboration and the flexibility to adapt to varying resource constraints and dynamic scenarios. Third, connectivity- and interaction-based intelligence emergence is a novel paradigm of AI Flow. By leveraging communication networks to enhance connectivity, the collaboration among AI models across heterogeneous nodes achieves emergent intelligence that surpasses the capability of any single model. The innovations of AI Flow provide enhanced intelligence, timely responsiveness, and ubiquitous accessibility to AI services, paving the way for the tighter fusion of AI techniques and communication systems.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

We study sampling algorithms for β-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of β-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size N mixes in O(log(N)).

We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.

Many problems in machine learning are naturally expressed in the language of undirected graphical models. Here, we propose black-box learning and inference algorithms for undirected models that optimize a variational approximation to the log-likelihood of the model. Central to our approach is an upper bound on the log-partition function parametrized by a function q that we express as a flexible neural network. Our bound makes it possible to track the partition function during learning, to speed-up sampling, and to train a broad class of hybrid directed/undirected models via a unified variational inference framework. We empirically demonstrate the effectiveness of our method on several popular generative modeling datasets.

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis alpha-entropy, depending on the value of the parameter.

The democratization of machine learning systems has made the process of fine-tuning accessible to practitioners, leading to a wide range of open-source models fine-tuned on specialized tasks and datasets. Recent work has proposed to merge such models to combine their functionalities. However, prior approaches are usually restricted to models that are fine-tuned from the same base model. Furthermore, the final merged model is typically required to be of the same size as the original models. In this work, we propose a new two-step algorithm to merge models -- termed PLeaS -- which relaxes these constraints. First, leveraging the Permutation symmetries inherent in the two models, PLeaS partially matches nodes in each layer by maximizing alignment. Next, PLeaS computes the weights of the merged model as a layer-wise Least Squares solution to minimize the approximation error between the features of the merged model and the permuted features of the original models. PLeaS allows a practitioner to merge two models sharing the same architecture into a single performant model of a desired size, even when the two original models are fine-tuned from different base models. We also demonstrate how our method can be extended to address a challenging scenario where no data is available from the fine-tuning domains. We demonstrate our method to merge ResNet and ViT models trained with shared and different label spaces, and show improvement over the state-of-the-art merging methods of up to 15 percentage points for the same target compute while merging models trained on DomainNet and fine-grained classification tasks. Our code is open-sourced at https://github.com/SewoongLab/PLeaS-Merging .

Building on the probability-domain distillation framework of Sparse-KD, we develop an axiomatic, operator-theoretic framework for multi-teacher ensemble knowledge distillation. Rather than prescribing a specific aggregation formula, we define five core axioms governing valid knowledge aggregation operators, encompassing convexity, positivity, continuity, weight monotonicity, and temperature coherence. We prove the existence and non-uniqueness of operator families satisfying these axioms, establishing that multiple distinct aggregation mechanisms conform to the same foundational principles. Within this framework, we establish operator-agnostic guarantees showing that multi-teacher aggregation reduces both stochastic variance and systematic supervisory bias under heterogeneous teachers, while providing Jensen-type bounds, log-loss guarantees, and safety attenuation properties. For aggregation operators linear in teacher weights, we further establish classical ensemble variance-reduction results under standard independence assumptions, with extensions to correlated-error regimes. The framework provides theoretical grounding for multi-teacher distillation from diverse frontier models while admitting multiple valid implementation strategies.

State space models (SSM) have recently been shown to be very effective as a deep learning layer as a promising alternative to sequence models such as RNNs, CNNs, or Transformers. The first version to show this potential was the S4 model, which is particularly effective on tasks involving long-range dependencies by using a prescribed state matrix called the HiPPO matrix. While this has an interpretable mathematical mechanism for modeling long dependencies, it introduces a custom representation and algorithm that can be difficult to implement. On the other hand, a recent variant of S4 called DSS showed that restricting the state matrix to be fully diagonal can still preserve the performance of the original model when using a specific initialization based on approximating S4's matrix. This work seeks to systematically understand how to parameterize and initialize such diagonal state space models. While it follows from classical results that almost all SSMs have an equivalent diagonal form, we show that the initialization is critical for performance. We explain why DSS works mathematically, by showing that the diagonal restriction of S4's matrix surprisingly recovers the same kernel in the limit of infinite state dimension. We also systematically describe various design choices in parameterizing and computing diagonal SSMs, and perform a controlled empirical study ablating the effects of these choices. Our final model S4D is a simple diagonal version of S4 whose kernel computation requires just 2 lines of code and performs comparably to S4 in almost all settings, with state-of-the-art results for image, audio, and medical time-series domains, and averaging 85\% on the Long Range Arena benchmark.

Distribution Matching Distillation (DMD) distills score-based generative models into efficient one-step generators, without requiring a one-to-one correspondence with the sampling trajectories of their teachers. However, limited model capacity causes one-step distilled models underperform on complex generative tasks, e.g., synthesizing intricate object motions in text-to-video generation. Directly extending DMD to multi-step distillation increases memory usage and computational depth, leading to instability and reduced efficiency. While prior works propose stochastic gradient truncation as a potential solution, we observe that it substantially reduces the generation diversity of multi-step distilled models, bringing it down to the level of their one-step counterparts. To address these limitations, we propose Phased DMD, a multi-step distillation framework that bridges the idea of phase-wise distillation with Mixture-of-Experts (MoE), reducing learning difficulty while enhancing model capacity. Phased DMD is built upon two key ideas: progressive distribution matching and score matching within subintervals. First, our model divides the SNR range into subintervals, progressively refining the model to higher SNR levels, to better capture complex distributions. Next, to ensure the training objective within each subinterval is accurate, we have conducted rigorous mathematical derivations. We validate Phased DMD by distilling state-of-the-art image and video generation models, including Qwen-Image (20B parameters) and Wan2.2 (28B parameters). Experimental results demonstrate that Phased DMD preserves output diversity better than DMD while retaining key generative capabilities. We will release our code and models.

The success of deep learning is due in large part to our ability to solve certain massive non-convex optimization problems with relative ease. Though non-convex optimization is NP-hard, simple algorithms -- often variants of stochastic gradient descent -- exhibit surprising effectiveness in fitting large neural networks in practice. We argue that neural network loss landscapes often contain (nearly) a single basin after accounting for all possible permutation symmetries of hidden units a la Entezari et al. 2021. We introduce three algorithms to permute the units of one model to bring them into alignment with a reference model in order to merge the two models in weight space. This transformation produces a functionally equivalent set of weights that lie in an approximately convex basin near the reference model. Experimentally, we demonstrate the single basin phenomenon across a variety of model architectures and datasets, including the first (to our knowledge) demonstration of zero-barrier linear mode connectivity between independently trained ResNet models on CIFAR-10. Additionally, we identify intriguing phenomena relating model width and training time to mode connectivity. Finally, we discuss shortcomings of the linear mode connectivity hypothesis, including a counterexample to the single basin theory.

For long-range percolation on Z with translation-invariant edge kernel J, it is a classical theorem of Aizenman and Newman (1986) that the phase transition is discontinuous when J(x,y) is of order |x-y|^{-2} and that there is no phase transition at all when J(x,y)=o(|x-y|^{-2}). We prove a strengthened version of this theorem for the hierarchical lattice, where the relevant threshold is at |x-y|^{-2d} loglog |x-y| rather than |x-y|^{-2}: There is a continuous phase transition for kernels of larger order, a discontinuous phase transition for kernels of exactly this order, and no phase transition at all for kernels of smaller order. As such, |x-y|^{-2d} loglog |x-y| is essentially the only kernel that produces a discontinuous phase transition. We also prove a hierarchical analogue of the ``M^2β=1'' conjecture of Imbrie and Newman (1988), which gives an exact formula for the density of the infinite cluster at the point of discontinuous phase transition and remains open in the Euclidean setting.

We present ModelTables, a benchmark of tables in Model Lakes that captures the structured semantics of performance and configuration tables often overlooked by text only retrieval. The corpus is built from Hugging Face model cards, GitHub READMEs, and referenced papers, linking each table to its surrounding model and publication context. Compared with open data lake tables, model tables are smaller yet exhibit denser inter table relationships, reflecting tightly coupled model and benchmark evolution. The current release covers over 60K models and 90K tables. To evaluate model and table relatedness, we construct a multi source ground truth using three complementary signals: (1) paper citation links, (2) explicit model card links and inheritance, and (3) shared training datasets. We present one extensive empirical use case for the benchmark which is table search. We compare canonical Data Lake search operators (unionable, joinable, keyword) and Information Retrieval baselines (dense, sparse, hybrid retrieval) on this benchmark. Union based semantic table retrieval attains 54.8 % P@1 overall (54.6 % on citation, 31.3 % on inheritance, 30.6 % on shared dataset signals); table based dense retrieval reaches 66.5 % P@1, and metadata hybrid retrieval achieves 54.1 %. This evaluation indicates clear room for developing better table search methods. By releasing ModelTables and its creation protocol, we provide the first large scale benchmark of structured data describing AI model. Our use case of table discovery in Model Lakes, provides intuition and evidence for developing more accurate semantic retrieval, structured comparison, and principled organization of structured model knowledge. Source code, data, and other artifacts have been made available at https://github.com/RJMillerLab/ModelTables.

Language models and specialized table embedding models have recently demonstrated strong performance on many tasks over tabular data. Researchers and practitioners are keen to leverage these models in many new application contexts; but limited understanding of the strengths and weaknesses of these models, and the table representations they generate, makes the process of finding a suitable model for a given task reliant on trial and error. There is an urgent need to gain a comprehensive understanding of these models to minimize inefficiency and failures in downstream usage. To address this need, we propose Observatory, a formal framework to systematically analyze embedding representations of relational tables. Motivated both by invariants of the relational data model and by statistical considerations regarding data distributions, we define eight primitive properties, and corresponding measures to quantitatively characterize table embeddings for these properties. Based on these properties, we define an extensible framework to evaluate language and table embedding models. We collect and synthesize a suite of datasets and use Observatory to analyze nine such models. Our analysis provides insights into the strengths and weaknesses of learned representations over tables. We find, for example, that some models are sensitive to table structure such as column order, that functional dependencies are rarely reflected in embeddings, and that specialized table embedding models have relatively lower sample fidelity. Such insights help researchers and practitioners better anticipate model behaviors and select appropriate models for their downstream tasks, while guiding researchers in the development of new models.

We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation through a Dyck tiling on a ribbon. We introduce a new combinatorial object similar to a tree-like tableau, which we call a forest. A forest is shown to give a permutation, and be bijective to a network corresponding to the inverse of the permutation. We show that the poset of networks is a finite graded lattice and admits an EL-labeling. By use of this EL-labeling, we show the lattice is supersolvable and compute the M\"obius function of an interval of the poset.

Clustering is a widely deployed unsupervised learning tool. Model-based clustering is a flexible framework to tackle data heterogeneity when the clusters have different shapes. Likelihood-based inference for mixture distributions often involves non-convex and high-dimensional objective functions, imposing difficult computational and statistical challenges. The classic expectation-maximization (EM) algorithm is a computationally thrifty iterative method that maximizes a surrogate function minorizing the log-likelihood of observed data in each iteration, which however suffers from bad local maxima even in the special case of the standard Gaussian mixture model with common isotropic covariance matrices. On the other hand, recent studies reveal that the unique global solution of a semidefinite programming (SDP) relaxed K-means achieves the information-theoretically sharp threshold for perfectly recovering the cluster labels under the standard Gaussian mixture model. In this paper, we extend the SDP approach to a general setting by integrating cluster labels as model parameters and propose an iterative likelihood adjusted SDP (iLA-SDP) method that directly maximizes the exact observed likelihood in the presence of data heterogeneity. By lifting the cluster assignment to group-specific membership matrices, iLA-SDP avoids centroids estimation -- a key feature that allows exact recovery under well-separateness of centroids without being trapped by their adversarial configurations. Thus iLA-SDP is less sensitive than EM to initialization and more stable on high-dimensional data. Our numeric experiments demonstrate that iLA-SDP can achieve lower mis-clustering errors over several widely used clustering methods including K-means, SDP and EM algorithms.

While foundation models update slowly due to resource-intensive training requirements, domain-specific models evolve between updates. Model merging aims to combine multiple expert models into a single, more capable model, thereby reducing storage and serving costs while supporting decentralized model development. Despite its potential, previous studies have primarily focused on merging visual classification models or Large Language Models (LLMs) for code and math tasks. Multimodal Large Language Models (MLLMs), which extend the capabilities of LLMs through large-scale multimodal training, have gained traction. However, there lacks a benchmark for model merging research that clearly divides the tasks for MLLM training and evaluation. In this paper, (i) we introduce the model merging benchmark for MLLMs, which includes multiple tasks such as VQA, Geometry, Chart, OCR, and Grounding, providing both LoRA and full fine-tuning models. Moreover, we explore how model merging can combine different modalities (e.g., vision-language, audio-language, and video-language models), moving toward the Omni-language model. (ii) We implement 10 model merging algorithms on the benchmark. Furthermore, we propose a novel method that removes noise from task vectors and robustly optimizes the merged vector based on a loss defined over task vector interactions, achieving an average performance gain of 2.48%. (iii) We find that model merging offers a promising way for building improved MLLMs without requiring data training. Our results also demonstrate that the complementarity among multiple modalities outperforms individual modalities.

Gross, Mansour, and Tucker introduced the partial-duality polynomial of a ribbon graph [Distributions, European J. Combin. 86, 1--20, 2020], the generating function enumerating partial duals by the Euler genus. Chmutov and Vignes-Tourneret wondered if this polynomial and its conjectured properties would hold for general delta-matroids, which are combinatorial abstractions of ribbon graphs. Yan and Jin contributed to this inquiry by identifying a subset of delta-matroids-specifically, even normal binary ones-whose twist polynomials are characterized by a singular term. Building upon this foundation, the current paper expands the scope of the investigation to encompass even non-binary delta-matroids, revealing that none of them have width-changing twists.

When considering a model architecture, there are several ways to reduce its memory footprint. Historically, popular approaches included selecting smaller architectures and creating sparse networks through pruning. More recently, randomized parameter-sharing (RPS) methods have gained traction for model compression at start of training. In this paper, we comprehensively assess the trade-off between memory and accuracy across RPS, pruning techniques, and building smaller models. Our findings demonstrate that RPS, which is both data and model-agnostic, consistently outperforms/matches smaller models and all moderately informed pruning strategies, such as MAG, SNIP, SYNFLOW, and GRASP, across the entire compression range. This advantage becomes particularly pronounced in higher compression scenarios. Notably, even when compared to highly informed pruning techniques like Lottery Ticket Rewinding (LTR), RPS exhibits superior performance in high compression settings. This points out inherent capacity advantage that RPS enjoys over sparse models. Theoretically, we establish RPS as a superior technique in terms of memory-efficient representation when compared to pruning for linear models. This paper argues in favor of paradigm shift towards RPS based models. During our rigorous evaluation of RPS, we identified issues in the state-of-the-art RPS technique ROAST, specifically regarding stability (ROAST's sensitivity to initialization hyperparameters, often leading to divergence) and Pareto-continuity (ROAST's inability to recover the accuracy of the original model at zero compression). We provably address both of these issues. We refer to the modified RPS, which incorporates our improvements, as STABLE-RPS.

Model merging aims to combine multiple expert models into a more capable single model, offering benefits such as reduced storage and serving costs, improved generalization, and support for decentralized model development. Despite its promise, previous studies have primarily focused on merging a few small models. This leaves many unanswered questions about the effect of scaling model size and how it interplays with other key factors -- like the base model quality and number of expert models -- , to affect the merged model's performance. This work systematically evaluates the utility of model merging at scale, examining the impact of these different factors. We experiment with merging fully fine-tuned models using 4 popular merging methods -- Averaging, Task~Arithmetic, Dare, and TIES -- across model sizes ranging from 1B-64B parameters and merging up to 8 different expert models. We evaluate the merged models on both held-in tasks, i.e., the expert's training tasks, and zero-shot generalization to unseen held-out tasks. Our experiments provide several new insights about model merging at scale and the interplay between different factors. First, we find that merging is more effective when experts are created from strong base models, i.e., models with good zero-shot performance. Second, larger models facilitate easier merging. Third merging consistently improves generalization capabilities. Notably, when merging 8 large expert models, the merged models often generalize better compared to the multitask trained models. Fourth, we can better merge more expert models when working with larger models. Fifth, different merging methods behave very similarly at larger scales. Overall, our findings shed light on some interesting properties of model merging while also highlighting some limitations. We hope that this study will serve as a reference point on large-scale merging for upcoming research.

The Platonic Representation Hypothesis suggests that independently trained neural networks converge to increasingly similar latent spaces. However, current strategies for mapping these representations are inherently pairwise, scaling quadratically with the number of models and failing to yield a consistent global reference. In this paper, we study the alignment of M ge 3 models. We first adapt Generalized Procrustes Analysis (GPA) to construct a shared orthogonal universe that preserves the internal geometry essential for tasks like model stitching. We then show that strict isometric alignment is suboptimal for retrieval, where agreement-maximizing methods like Canonical Correlation Analysis (CCA) typically prevail. To bridge this gap, we finally propose Geometry-Corrected Procrustes Alignment (GCPA), which establishes a robust GPA-based universe followed by a post-hoc correction for directional mismatch. Extensive experiments demonstrate that GCPA consistently improves any-to-any retrieval while retaining a practical shared reference space.

Model merging has emerged as a promising technique for combining multiple fine-tuned models into a single multitask model without retraining. However, the factors that determine whether merging will succeed or fail remain poorly understood. In this work, we investigate why specific models are merged better than others. To do so, we propose a concrete, measurable definition of mergeability. We investigate several potential causes for high or low mergeability, highlighting the base model knowledge as a dominant factor: Models fine-tuned on instances that the base model knows better are more mergeable than models fine-tuned on instances that the base model struggles with. Based on our mergeability definition, we explore a simple weighted merging technique that better preserves weak knowledge in the base model.

Mechanistic Interpretability aims to reverse engineer the algorithms implemented by neural networks by studying their weights and activations. An obstacle to reverse engineering neural networks is that many of the parameters inside a network are not involved in the computation being implemented by the network. These degenerate parameters may obfuscate internal structure. Singular learning theory teaches us that neural network parameterizations are biased towards being more degenerate, and parameterizations with more degeneracy are likely to generalize further. We identify 3 ways that network parameters can be degenerate: linear dependence between activations in a layer; linear dependence between gradients passed back to a layer; ReLUs which fire on the same subset of datapoints. We also present a heuristic argument that modular networks are likely to be more degenerate, and we develop a metric for identifying modules in a network that is based on this argument. We propose that if we can represent a neural network in a way that is invariant to reparameterizations that exploit the degeneracies, then this representation is likely to be more interpretable, and we provide some evidence that such a representation is likely to have sparser interactions. We introduce the Interaction Basis, a tractable technique to obtain a representation that is invariant to degeneracies from linear dependence of activations or Jacobians.

We study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graphs, in terms of cliques which are intersections of balls. We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.

Merging Large Language Models (LLMs) is a cost-effective technique for combining multiple expert LLMs into a single versatile model, retaining the expertise of the original ones. However, current approaches often overlook the importance of safety alignment during merging, leading to highly misaligned models. This work investigates the effects of model merging on alignment. We evaluate several popular model merging techniques, demonstrating that existing methods do not only transfer domain expertise but also propagate misalignment. We propose a simple two-step approach to address this problem: (i) generating synthetic safety and domain-specific data, and (ii) incorporating these generated data into the optimization process of existing data-aware model merging techniques. This allows us to treat alignment as a skill that can be maximized in the resulting merged LLM. Our experiments illustrate the effectiveness of integrating alignment-related data during merging, resulting in models that excel in both domain expertise and alignment.

The advancement of Large Language Models (LLMs) for domain applications in fields such as materials science and engineering depends on the development of fine-tuning strategies that adapt models for specialized, technical capabilities. In this work, we explore the effects of Continued Pretraining (CPT), Supervised Fine-Tuning (SFT), and various preference-based optimization approaches, including Direct Preference Optimization (DPO) and Odds Ratio Preference Optimization (ORPO), on fine-tuned LLM performance. Our analysis shows how these strategies influence model outcomes and reveals that the merging of multiple fine-tuned models can lead to the emergence of capabilities that surpass the individual contributions of the parent models. We find that model merging leads to new functionalities that neither parent model could achieve alone, leading to improved performance in domain-specific assessments. Experiments with different model architectures are presented, including Llama 3.1 8B and Mistral 7B models, where similar behaviors are observed. Exploring whether the results hold also for much smaller models, we use a tiny LLM with 1.7 billion parameters and show that very small LLMs do not necessarily feature emergent capabilities under model merging, suggesting that model scaling may be a key component. In open-ended yet consistent chat conversations between a human and AI models, our assessment reveals detailed insights into how different model variants perform and show that the smallest model achieves a high intelligence score across key criteria including reasoning depth, creativity, clarity, and quantitative precision. Other experiments include the development of image generation prompts based on disparate biological material design concepts, to create new microstructures, architectural concepts, and urban design based on biological materials-inspired construction principles.

In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical foundation for machine learning in p-adic metric spaces, which naturally respect hierarchical structure. Our main result proves that an n-dimensional plane minimizing the p-adic sum of distances to points in a dataset must pass through at least n + 1 of those points -- a striking contrast to Euclidean regression that highlights how p-adic metrics better align with the discrete nature of hierarchical data. As a corollary, a polynomial of degree n constructed to minimise the p-adic sum of residuals will pass through at least n + 1 points. As a further corollary, a polynomial of degree n approximating a higher degree polynomial at a finite number of points will yield a difference polynomial that has distinct rational roots. We demonstrate the practical significance of this result through two applications in natural language processing: analyzing hierarchical taxonomies and modeling grammatical morphology. These results suggest that p-adic metrics may be fundamental to properly handling hierarchical data structures in machine learning. In hierarchical data, interpolation between points often makes less sense than selecting actual observed points as representatives.

Model merging is an efficient way of obtaining a multi-task model from several pretrained models without further fine-tuning, and it has gained attention in various domains, including natural language processing (NLP). Despite the efficiency, a key challenge in model merging is the seemingly inevitable decrease in task performance as the number of models increases. In this paper, we propose Spectral Truncation And Rescale (STAR) that aims at mitigating ``merging conflicts'' by truncating small components in the respective spectral spaces, which is followed by an automatic parameter rescaling scheme to retain the nuclear norm of the original matrix. STAR requires no additional inference on original training data and is robust to hyperparamater choice. We demonstrate the effectiveness of STAR through extensive model merging cases on diverse NLP tasks. Specifically, STAR works robustly across varying model sizes, and can outperform baselines by 4.2% when merging 12 models on Flan-T5. Our code is publicly available at https://github.com/IBM/STAR.

Although disentangled representations are often said to be beneficial for downstream tasks, current empirical and theoretical understanding is limited. In this work, we provide evidence that disentangled representations coupled with sparse base-predictors improve generalization. In the context of multi-task learning, we prove a new identifiability result that provides conditions under which maximally sparse base-predictors yield disentangled representations. Motivated by this theoretical result, we propose a practical approach to learn disentangled representations based on a sparsity-promoting bi-level optimization problem. Finally, we explore a meta-learning version of this algorithm based on group Lasso multiclass SVM base-predictors, for which we derive a tractable dual formulation. It obtains competitive results on standard few-shot classification benchmarks, while each task is using only a fraction of the learned representations.

Skid-steered wheel mobile robots (SSWMRs) are characterized by the unique domination of the tire-terrain skidding for the robot to move. The lack of reliable friction models cascade into unreliable motion models, especially the reduced ordered variants used for state estimation and robot control. Ensemble modeling is an emerging research direction where the overall motion model is broken down into a family of local models to distribute the performance and resource requirement and provide a fast real-time prediction. To this end, a gaussian mixture model based modeling identification of model clusters is adopted and implemented within an interactive multiple model (IMM) based state estimation. The framework is adopted and implemented for angular velocity as the estimated state for a mid scaled skid-steered wheel mobile robot platform.

Given a set of K probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Model merging combines multiple fine-tuned checkpoints into a single model without additional training, offering an attractive approach to reusing models and efficiently improving performance. However, it remains unclear whether the advantages reported for settings where all merged experts have distinct roles and are tuned on clearly separated tasks also hold in settings where the merged experts do not have clearly distinct roles, but are trained on overlapping or even conflicting objectives. To evaluate this setting, we present a large-scale, systematic evaluation of "in-the-wild" model merging of heterogeneous experts, that may have been trained on overlapping or conflicting objectives. Concretely, we evaluate six state-of-the-art merging methods, including recent subspace methods, across four open-weight LLMs, twelve fine-tuned checkpoints per base model, and sixteen standard LLM benchmarks. Evaluating through standardized benchmarks, we measure both the probability that a model merged from a heterogeneous set of experts outperforms the base model and we measure relative gains over the best individual checkpoint. Our results show that the oldest and simplest method, Task Arithmetic, is the only approach that reliably yields performance gains on LLMs in this "in-the-wild" setting. Other interference-aware and subspace merging methods typically do not result in notable improvements over the base model. Our findings indicate that current merging techniques mostly do not enable extracting useful weight updates from heterogeneous and potentially conflicting versions. This motivates the design of LLM-specific merging algorithms and merging-aware fine-tuning methods.

Understanding how language model performance varies with scale is critical to benchmark and algorithm development. Scaling laws are one approach to building this understanding, but the requirement of training models across many different scales has limited their use. We propose an alternative, observational approach that bypasses model training and instead builds scaling laws from ~80 publically available models. Building a single scaling law from multiple model families is challenging due to large variations in their training compute efficiencies and capabilities. However, we show that these variations are consistent with a simple, generalized scaling law where language model performance is a function of a low-dimensional capability space, and model families only vary in their efficiency in converting training compute to capabilities. Using this approach, we show the surprising predictability of complex scaling phenomena: we show that several emergent phenomena follow a smooth, sigmoidal behavior and are predictable from small models; we show that the agent performance of models such as GPT-4 can be precisely predicted from simpler non-agentic benchmarks; and we show how to predict the impact of post-training interventions like Chain-of-Thought and Self-Consistency as language model capabilities continue to improve.

The precise equivalence between discretized Euclidean field theories and a certain class of probabilistic graphical models, namely the mathematical framework of Markov random fields, opens up the opportunity to investigate machine learning from the perspective of quantum field theory. In this contribution we will demonstrate, through the Hammersley-Clifford theorem, that the φ^{4} scalar field theory on a square lattice satisfies the local Markov property and can therefore be recast as a Markov random field. We will then derive from the φ^{4} theory machine learning algorithms and neural networks which can be viewed as generalizations of conventional neural network architectures. Finally, we will conclude by presenting applications based on the minimization of an asymmetric distance between the probability distribution of the φ^{4} machine learning algorithms and target probability distributions.

The surprising ability of Large Language Models (LLMs) to perform well on complex reasoning with only few-shot chain-of-thought prompts is believed to emerge only in very large-scale models (100+ billion parameters). We show that such abilities can, in fact, be distilled down from GPT-3.5 (ge 175B) to T5 variants (le 11B). We propose model specialization, to specialize the model's ability towards a target task. The hypothesis is that large models (commonly viewed as larger than 100B) have strong modeling power, but are spread on a large spectrum of tasks. Small models (commonly viewed as smaller than 10B) have limited model capacity, but if we concentrate their capacity on a specific target task, the model can achieve a decent improved performance. We use multi-step math reasoning as our testbed because it is a very typical emergent ability. We show two important aspects of model abilities: (1). there exists a very complex balance/ tradeoff between language models' multi-dimensional abilities; (2). by paying the price of decreased generic ability, we can clearly lift up the scaling curve of models smaller than 10B towards a specialized multi-step math reasoning ability. We further give comprehensive discussions about important design choices for better generalization, including the tuning data format, the start model checkpoint, and a new model selection method. We hope our practice and discoveries can serve as an important attempt towards specialized smaller models in the new research paradigm set by LLMs.

State Space Models (SSMs) have emerged as a promising alternative to the popular transformer-based models and have been increasingly gaining attention. Compared to transformers, SSMs excel at tasks with sequential data or longer contexts, demonstrating comparable performances with significant efficiency gains. In this survey, we provide a coherent and systematic overview for SSMs, including their theoretical motivations, mathematical formulations, comparison with existing model classes, and various applications. We divide the SSM series into three main sections, providing a detailed introduction to the original SSM, the structured SSM represented by S4, and the selective SSM typified by Mamba. We put an emphasis on technicality, and highlight the various key techniques introduced to address the effectiveness and efficiency of SSMs. We hope this manuscript serves as an introduction for researchers to explore the theoretical foundations of SSMs.

Using the weights of trained Neural Network (NN) models as data modality has recently gained traction as a research field - dubbed Weight Space Learning (WSL). Multiple recent works propose WSL methods to analyze models, evaluate methods, or synthesize weights. Weight space learning methods require populations of trained models as datasets for development and evaluation. However, existing collections of models - called `model zoos' - are unstructured or follow a rudimentary definition of diversity. In parallel, work rooted in statistical physics has identified phases and phase transitions in NN models. Models are homogeneous within the same phase but qualitatively differ from one phase to another. We combine the idea of `model zoos' with phase information to create a controlled notion of diversity in populations. We introduce 12 large-scale zoos that systematically cover known phases and vary over model architecture, size, and datasets. These datasets cover different modalities, such as computer vision, natural language processing, and scientific ML. For every model, we compute loss landscape metrics and validate full coverage of the phases. With this dataset, we provide the community with a resource with a wide range of potential applications for WSL and beyond. Evidence suggests the loss landscape phase plays a role in applications such as model training, analysis, or sparsification. We demonstrate this in an exploratory study of the downstream methods like transfer learning or model weights averaging.

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular low displacement rank) for integrating tensor fields defined on weighted trees. Several applications of the resulting fast tree-field integrators (FTFIs) are presented, including (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) Topological Transformers (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as three extra learnable parameters per Transformer layer, leading to 1.0-1.5%+ accuracy gains. Importantly, most of FTFIs are exact methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide 5.7-13x speedups. We also provide an extensive theoretical analysis of our methods.

Foundation multi-modal models are often designed by stitching of multiple existing pretrained uni-modal models: for example, an image classifier with an text model. This stitching process is performed by training a connector module that aims to align the representation spaces of these uni-modal models towards a multi-modal objective. However, given the complexity of training such connectors on large scale web-based datasets coupled with the ever-increasing number of available pretrained uni-modal models, the task of uni-modal models selection and subsequent connector module training becomes computationally demanding. To address this under-studied critical problem, we propose Hypernetwork Model Alignment (Hyma), a novel all-in-one solution for optimal uni-modal model selection and connector training by leveraging hypernetworks. Specifically, our framework utilizes the parameter prediction capability of a hypernetwork to obtain jointly trained connector modules for N times M combinations of uni-modal models. In our experiments, Hyma reduces the cost of searching for the best performing uni-modal model pair by 10times, while matching the ranking and trained connector performance obtained via grid search across a suite of diverse multi-modal benchmarks.

As Large Language Models (LLMs) excel across tasks and specialized domains, scaling LLMs based on existing models has garnered significant attention, which faces the challenge of decreasing performance when combining disparate models. Various techniques have been proposed for the aggregation of pre-trained LLMs, including model merging, Mixture-of-Experts, and stacking. Despite their merits, a comprehensive comparison and synergistic application of them to a diverse model zoo is yet to be adequately addressed. In light of this research gap, this paper introduces Model-GLUE, a holistic LLM scaling guideline. First, our work starts with a benchmarking of existing LLM scaling techniques, especially selective merging, and variants of mixture. Utilizing the insights from the benchmark results, we formulate an strategy for the selection and aggregation of a heterogeneous model zoo characterizing different architectures and initialization. Our methodology involves the clustering of mergeable models and optimal merging strategy selection, and the integration of clusters through a model mixture. Finally, evidenced by our experiments on a diverse Llama-2-based model zoo, Model-GLUE shows an average performance enhancement of 5.61%, achieved without additional training. Codes are available at: https://github.com/Model-GLUE/Model-GLUE.

Hyper-Connections (HC) generalize residual connections into multiple streams, employing residual matrices for cross-stream feature mixing to enrich model expressivity. However, unconstrained mixing disrupts the identity mapping property intrinsic to the residual connection, causing unstable training. To address this, Manifold-Constrained Hyper-Connections (mHC) and its variant restrict these matrices to the Birkhoff polytope (doubly stochastic matrices) via Sinkhorn iterations or permutation-based parameterizations. We reveal three limitations of this polytope constraint: (1) identity degeneration, where learned matrices collapse around the identity and diminish cross-stream interactions, (2) an expressivity bottleneck, as the non-negativity constraint prevents subtractive feature disentanglement, and (3) parameterization inefficiencies, manifesting as unstable Sinkhorn iterations or the factorial-scaling overhead of permutation-based parameterizations. To overcome these flaws, we propose Spectral-Sphere-Constrained Hyper-Connections (sHC). By geometrically shifting the feasible set from a rigid polytope to a spectral norm sphere, sHC allows negative entries, unlocking subtractive interactions for selective feature diversification. This shift eliminates unstable Sinkhorn projections and factorial parameterization, enabling expressive, non-degenerate residual matrices while preserving training stability.

Deep model merging represents an emerging research direction that combines multiple fine-tuned models to harness their specialized capabilities across different tasks and domains. Current model merging techniques focus on merging all available models simultaneously, with weight interpolation-based methods being the predominant approaches. However, these conventional approaches are not well-suited for scenarios where models become available sequentially, and they often suffer from high memory requirements and potential interference between tasks. In this study, we propose a training-free projection-based continual merging method that processes models sequentially through orthogonal projections of weight matrices and adaptive scaling mechanisms. Our method operates by projecting new parameter updates onto subspaces orthogonal to existing merged parameter updates while using an adaptive scaling mechanism to maintain stable parameter distances, enabling efficient sequential integration of task-specific knowledge. Our approach maintains constant memory complexity to the number of models, minimizes interference between tasks through orthogonal projections, and retains the performance of previously merged models through adaptive task vector scaling. Extensive experiments on CLIP-ViT models demonstrate that our method achieves a 5-8% average accuracy improvement while maintaining robust performance in different task orderings.

Protein family design emerges as a promising alternative by combining the advantages of de novo protein design and mutation-based directed evolution.In this paper, we propose ProfileBFN, the Profile Bayesian Flow Networks, for specifically generative modeling of protein families. ProfileBFN extends the discrete Bayesian Flow Network from an MSA profile perspective, which can be trained on single protein sequences by regarding it as a degenerate profile, thereby achieving efficient protein family design by avoiding large-scale MSA data construction and training. Empirical results show that ProfileBFN has a profound understanding of proteins. When generating diverse and novel family proteins, it can accurately capture the structural characteristics of the family. The enzyme produced by this method is more likely than the previous approach to have the corresponding function, offering better odds of generating diverse proteins with the desired functionality.

In the era of large language models, model merging is a promising way to combine multiple task-specific models into a single multitask model without extra training. However, two challenges remain: (a) interference between different models and (b) heterogeneous data during testing. Traditional model merging methods often show significant performance gaps compared to fine-tuned models due to these issues. Additionally, a one-size-fits-all model lacks flexibility for diverse test data, leading to performance degradation. We show that both shared and exclusive task-specific knowledge are crucial for merging performance, but directly merging exclusive knowledge hinders overall performance. In view of this, we propose Twin-Merging, a method that encompasses two principal stages: (1) modularizing knowledge into shared and exclusive components, with compression to reduce redundancy and enhance efficiency; (2) dynamically merging shared and task-specific knowledge based on the input. This approach narrows the performance gap between merged and fine-tuned models and improves adaptability to heterogeneous data. Extensive experiments on 12 datasets for both discriminative and generative tasks demonstrate the effectiveness of our method, showing an average improvement of 28.34% in absolute normalized score for discriminative tasks and even surpassing the fine-tuned upper bound on the generative tasks. (Our implementation is available in https://github.com/LZY-the-boys/Twin-Mergin.)

We prove that the celebrated Planar Product Structure Theorem by Dujmovic et al, and also related graph product structure results, can be formulated with the induced subgraph containment relation. Precisely, we prove that if a graph G is a subgraph of the strong product of a graph Q of bounded maximum degree (such as a path) and a graph M of bounded tree-width, then G is an induced subgraph of the strong product of Q and a graph M' of bounded tree-width being at most exponential in the maximum degree of Q and the tree-width of M. In particular, if G is planar, we show that G is an induced subgraph of the strong product of a path and a graph of tree-width 39. In the course of proving this result, we introduce and study H-clique-width, a new single structural measure that captures a hereditary analogue of the traditional product structure (where, informally, the strong product has one factor from the graph class H and one factor of bounded clique-width).

Neural models are usually adapted through changes in parameters shared among model components via fine-tuning, alignment-based training, and reinforcement learning. These changes have been found effective in short-term optimization. However, they result in long-term alterations in the model's base behavior. In this study, we introduce the concept of structural irreversibility as a characteristic of shared-parameter model adaptation. This concept refers to the intertwining of task-specific objectives with the representational identity of the model. We show that when parameters are directly mutated, the resulting model behaves divergently from the original model. This divergence cannot be reversed deterministically without an explicit parameter snapshot. We introduce reversible behavioral learning, in which model behaviors are structurally dissociated from identity parameters and can be deterministically unloaded through an explicit unload process. We also introduce the Recoverability Factor as a normalized measure of behavioral recoverability and provide additional diagnostics based on model divergence. Experiments show that reversible model adaptation achieves rollback within numerical precision, whereas shared-parameter mutation exhibits persistent post-reset divergence.

Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.

Political scientists are rapidly adopting large language models (LLMs) for text annotation, yet the sensitivity of annotation results to implementation choices remains poorly understood. Most evaluations test a single model or configuration; how model choice, model size, learning approach, and prompt style interact, and whether popular "best practices" survive controlled comparison, are largely unexplored. We present a controlled evaluation of these pipeline choices, testing six open-weight models across four political science annotation tasks under identical quantisation, hardware, and prompt-template conditions. Our central finding is methodological: interaction effects dominate main effects, so seemingly reasonable pipeline choices can become consequential researcher degrees of freedom. No single model, prompt style, or learning approach is uniformly superior, and the best-performing model varies across tasks. Two corollaries follow. First, model size is an unreliable guide both to cost and to performance: cross-family efficiency differences are so large that some larger models are less resource-intensive than much smaller alternatives, while within model families mid-range variants often match or exceed larger counterparts. Second, widely recommended prompt engineering techniques yield inconsistent and sometimes negative effects on annotation performance. We use these benchmark results to develop a validation-first framework - with a principled ordering of pipeline decisions, guidance on prompt freezing and held-out evaluation, reporting standards, and open-source tools - to help researchers navigate this decision space transparently.

We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

In recent years, there has been a development in approaching rationality problems through the motivic methods (cf. [Kontsevich--Tschinkel'19], [Nicaise--Shinder'19], [Nicaise--Ottem'21]). This method requires the explicit construction of degeneration families of curves with favorable properties. While the specific construction is generally difficult, [Nicaise--Ottem'22] combines combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and shows the non-stable rationality of a very general hypersurface in projective spaces. In this paper, we extend the result of [Nicaise--Ottem'22] not only for hypersurfaces in algebraic tori but also to those in sch\"{o}n affine varieties. In application, we show the irrationality of certain hypersurfaces in the complex Grassmannian variety Gr(2, n) using the motivic method, which coincides with the result obtained by the same author in the previous research.

Model merging aims to cheaply combine individual task-specific models into a single multitask model. In this work, we view past merging methods as leveraging different notions of a ''task subspace'' in which models are matched before being merged. We connect the task subspace of a given model to its loss landscape and formalize how this approach to model merging can be seen as solving a linear system of equations. While past work has generally been limited to linear systems that have a closed-form solution, we consider using the conjugate gradient method to find a solution. We show that using the conjugate gradient method can outperform closed-form solutions, enables merging via linear systems that are otherwise intractable to solve, and flexibly allows choosing from a wide variety of initializations and estimates for the ''task subspace''. We ultimately demonstrate that our merging framework called ''Matching Models in their Task Subspace'' (MaTS) achieves state-of-the-art results in multitask and intermediate-task model merging. We release all of the code and checkpoints used in our work at https://github.com/r-three/mats.

We introduce marginalization models (MaMs), a new family of generative models for high-dimensional discrete data. They offer scalable and flexible generative modeling with tractable likelihoods by explicitly modeling all induced marginal distributions. Marginalization models enable fast evaluation of arbitrary marginal probabilities with a single forward pass of the neural network, which overcomes a major limitation of methods with exact marginal inference, such as autoregressive models (ARMs). We propose scalable methods for learning the marginals, grounded in the concept of "marginalization self-consistency". Unlike previous methods, MaMs support scalable training of any-order generative models for high-dimensional problems under the setting of energy-based training, where the goal is to match the learned distribution to a given desired probability (specified by an unnormalized (log) probability function such as energy function or reward function). We demonstrate the effectiveness of the proposed model on a variety of discrete data distributions, including binary images, language, physical systems, and molecules, for maximum likelihood and energy-based training settings. MaMs achieve orders of magnitude speedup in evaluating the marginal probabilities on both settings. For energy-based training tasks, MaMs enable any-order generative modeling of high-dimensional problems beyond the capability of previous methods. Code is at https://github.com/PrincetonLIPS/MaM.

The Automunge open source python library platform for tabular data pre-processing automates feature engineering data transformations of numerical encoding and missing data infill to received tidy data on bases fit to properties of columns in a designated train set for consistent and efficient application to subsequent data pipelines such as for inference, where transformations may be applied to distinct columns in "family tree" sets with generations and branches of derivations. Included in the library of transformations are methods to extract structure from bounded categorical string sets by way of automated string parsing, in which comparisons between entries in the set of unique values are parsed to identify character subset overlaps which may be encoded by appended columns of boolean overlap detection activations or by replacing string entries with identified overlap partitions. Further string parsing options, which may also be applied to unbounded categoric sets, include extraction of numeric substring partitions from entries or search functions to identify presence of specified substring partitions. The aggregation of these methods into "family tree" sets of transformations are demonstrated for use to automatically extract structure from categoric string compositions in relation to the set of entries in a column, such as may be applied to prepare categoric string set encodings for machine learning without human intervention.

Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular model classes: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose polynomial circuits as a suitable machine learning model. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values.