Daily Papers

Complex tables with multi-level headers, merged cells and heterogeneous layouts pose persistent challenges for LLMs in both understanding and reasoning. Existing approaches typically rely on table linearization or normalized grid modeling. However, these representations struggle to explicitly capture hierarchical structures and cross-dimensional dependencies, which can lead to misalignment between structural semantics and textual representations for non-standard tables. To address this issue, we propose an Orthogonal Hierarchical Decomposition (OHD) framework that constructs structure-preserving input representations of complex tables for LLMs. OHD introduces an Orthogonal Tree Induction (OTI) method based on spatial--semantic co-constraints, which decomposes irregular tables into a column tree and a row tree to capture vertical and horizontal hierarchical dependencies, respectively. Building on this representation, we design a dual-pathway association protocol to symmetrically reconstruct semantic lineage of each cell, and incorporate an LLM as a semantic arbitrator to align multi-level semantic information. We evaluate OHD framework on two complex table question answering benchmarks, AITQA and HiTab. Experimental results show that OHD consistently outperforms existing representation paradigms across multiple evaluation metrics.

Low-rank optimization has emerged as a promising direction in training large language models (LLMs) to reduce the memory usage of adaptive optimizers by constraining learning to a lower-dimensional space. Prior work typically projects gradients of linear layers using approaches based on Singular Value Decomposition (SVD). However, applying SVD-based procedures individually to each layer in large models is computationally expensive and incurs additional memory costs due to storing the projection matrices. In this work, we propose a computationally efficient and conceptually simple two-step procedure to approximate SVD-based gradient projections into lower-dimensional spaces. First, we construct a complete orthogonal basis using predefined orthogonal matrices of the Discrete Cosine Transform (DCT). Second, we adaptively select basis columns based on their alignment with the gradient of each layer. Each projection matrix in our method is obtained via a single matrix multiplication followed by a lightweight sorting step to identify the most relevant basis vectors. Due to the predefined nature of the orthogonal bases, they are computed once at the start of training. During training, we store only the indices of the selected columns, avoiding the need to store full projection matrices for each layer. Our numerical experiments on both pre-training and fine-tuning tasks demonstrate the effectiveness of our dual strategy in approximating optimal low-rank projections, matching the performance of costly SVD-based methods while achieving faster runtime and reduced memory usage.

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.

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.

Updating a truncated Singular Value Decomposition (SVD) is crucial in representation learning, especially when dealing with large-scale data matrices that continuously evolve in practical scenarios. Aligning SVD-based models with fast-paced updates becomes increasingly important. Existing methods for updating truncated SVDs employ Rayleigh-Ritz projection procedures, where projection matrices are augmented based on original singular vectors. However, these methods suffer from inefficiency due to the densification of the update matrix and the application of the projection to all singular vectors. To address these limitations, we introduce a novel method for dynamically approximating the truncated SVD of a sparse and temporally evolving matrix. Our approach leverages sparsity in the orthogonalization process of augmented matrices and utilizes an extended decomposition to independently store projections in the column space of singular vectors. Numerical experiments demonstrate a remarkable efficiency improvement of an order of magnitude compared to previous methods. Remarkably, this improvement is achieved while maintaining a comparable precision to existing approaches.

Learning decompositions of expensive-to-evaluate black-box functions promises to scale Bayesian optimisation (BO) to high-dimensional problems. However, the success of these techniques depends on finding proper decompositions that accurately represent the black-box. While previous works learn those decompositions based on data, we investigate data-independent decomposition sampling rules in this paper. We find that data-driven learners of decompositions can be easily misled towards local decompositions that do not hold globally across the search space. Then, we formally show that a random tree-based decomposition sampler exhibits favourable theoretical guarantees that effectively trade off maximal information gain and functional mismatch between the actual black-box and its surrogate as provided by the decomposition. Those results motivate the development of the random decomposition upper-confidence bound algorithm (RDUCB) that is straightforward to implement - (almost) plug-and-play - and, surprisingly, yields significant empirical gains compared to the previous state-of-the-art on a comprehensive set of benchmarks. We also confirm the plug-and-play nature of our modelling component by integrating our method with HEBO, showing improved practical gains in the highest dimensional tasks from Bayesmark.

We present an algorithm for decomposing low rank tensors of any symmetry type, from fully asymmetric to fully symmetric. It generalizes the recent subspace power method from symmetric tensors to all tensors. The algorithm transforms an input tensor into a tensor with orthonormal slices. We show that for tensors with orthonormal slices and low rank, the summands of their decomposition are in one-to-one correspondence with the partially symmetric singular vector tuples (pSVTs) with singular value one. We use this to show correctness of the algorithm. We introduce a shifted power method for computing pSVTs and establish its global convergence. Numerical experiments demonstrate that our decomposition algorithm achieves higher accuracy and faster runtime than existing methods.

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.

Diffusion models, which learn to reverse a signal destruction process to generate new data, typically require the signal at each step to have the same dimension. We argue that, considering the spatial redundancy in image signals, there is no need to maintain a high dimensionality in the evolution process, especially in the early generation phase. To this end, we make a theoretical generalization of the forward diffusion process via signal decomposition. Concretely, we manage to decompose an image into multiple orthogonal components and control the attenuation of each component when perturbing the image. That way, along with the noise strength increasing, we are able to diminish those inconsequential components and thus use a lower-dimensional signal to represent the source, barely losing information. Such a reformulation allows to vary dimensions in both training and inference of diffusion models. Extensive experiments on a range of datasets suggest that our approach substantially reduces the computational cost and achieves on-par or even better synthesis performance compared to baseline methods. We also show that our strategy facilitates high-resolution image synthesis and improves FID of diffusion model trained on FFHQ at 1024times1024 resolution from 52.40 to 10.46. Code and models will be made publicly available.

In time series forecasting, effectively disentangling intricate temporal patterns is crucial. While recent works endeavor to combine decomposition techniques with deep learning, multiple frequencies may still be mixed in the decomposed components, e.g., trend and seasonal. Furthermore, frequency domain analysis methods, e.g., Fourier and wavelet transforms, have limitations in resolution in the time domain and adaptability. In this paper, we propose D-PAD, a deep-shallow multi-frequency patterns disentangling neural network for time series forecasting. Specifically, a multi-component decomposing (MCD) block is introduced to decompose the series into components with different frequency ranges, corresponding to the "shallow" aspect. A decomposition-reconstruction-decomposition (D-R-D) module is proposed to progressively extract the information of frequencies mixed in the components, corresponding to the "deep" aspect. After that, an interaction and fusion (IF) module is used to further analyze the components. Extensive experiments on seven real-world datasets demonstrate that D-PAD achieves the state-of-the-art performance, outperforming the best baseline by an average of 9.48% and 7.15% in MSE and MAE, respectively.

Residual connections are pivotal for deep neural networks, enabling greater depth by mitigating vanishing gradients. However, in standard residual updates, the module's output is directly added to the input stream. This can lead to updates that predominantly reinforce or modulate the existing stream direction, potentially underutilizing the module's capacity for learning entirely novel features. In this work, we introduce Orthogonal Residual Update: we decompose the module's output relative to the input stream and add only the component orthogonal to this stream. This design aims to guide modules to contribute primarily new representational directions, fostering richer feature learning while promoting more efficient training. We demonstrate that our orthogonal update strategy improves generalization accuracy and training stability across diverse architectures (ResNetV2, Vision Transformers) and datasets (CIFARs, TinyImageNet, ImageNet-1k), achieving, for instance, a +4.3\%p top-1 accuracy gain for ViT-B on ImageNet-1k.

Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy enables parameter sharing and efficient reconstruction and has been widely adopted across domains ranging from multi-view learning and signal processing to neural network compression. However, it leaves a fundamental question unanswered: which matrices can be safely concatenated and compressed together under explicit reconstruction error constraints? Existing approaches rely on heuristic or architecture-specific grouping and provide no principled guarantees on the resulting SVD approximation error. In the present work, we introduce a theory-driven framework for compression-aware clustering of matrices under SVD compression constraints. Our analysis establishes new spectral bounds for horizontally concatenated matrices, deriving global upper bounds on the optimal rank-r SVD reconstruction error from lower bounds on singular value growth. The first bound follows from Weyl-type monotonicity under blockwise extensions, while the second leverages singular values of incremental residuals to yield tighter, per-block guarantees. We further develop an efficient approximate estimator based on incremental truncated SVD that tracks dominant singular values without forming the full concatenated matrix. Therefore, we propose three clustering algorithms that merge matrices only when their predicted joint SVD compression error remains below a user-specified threshold. The algorithms span a trade-off between speed, provable accuracy, and scalability, enabling compression-aware clustering with explicit error control. Code is available online.

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models.

Dictionary learning is a widely used unsupervised learning method in signal processing and machine learning. Most existing works of dictionary learning are in an offline manner. There are mainly two offline ways for dictionary learning. One is to do an alternative optimization of both the dictionary and the sparse code; the other way is to optimize the dictionary by restricting it over the orthogonal group. The latter one is called orthogonal dictionary learning which has a lower complexity implementation, hence, it is more favorable for lowcost devices. However, existing schemes on orthogonal dictionary learning only work with batch data and can not be implemented online, which is not applicable for real-time applications. This paper proposes a novel online orthogonal dictionary scheme to dynamically learn the dictionary from streaming data without storing the historical data. The proposed scheme includes a novel problem formulation and an efficient online algorithm design with convergence analysis. In the problem formulation, we relax the orthogonal constraint to enable an efficient online algorithm. In the algorithm design, we propose a new Frank-Wolfe-based online algorithm with a convergence rate of O(ln t/t^(1/4)). The convergence rate in terms of key system parameters is also derived. Experiments with synthetic data and real-world sensor readings demonstrate the effectiveness and efficiency of the proposed online orthogonal dictionary learning scheme.

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore-Penrose inverse (which is consistent with respect to unitary/orthonormal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically-important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions and examples of their use are described.

Recently, triple decomposition has attracted increasing attention for decomposing third-order tensors into three factor tensors. However, this approach is limited to third-order tensors and enforces uniformity in the lower dimensions across all factor tensors, which restricts its flexibility and applicability. To address these issues, we propose the Multiple decomposition, a novel framework that generalizes triple decomposition to arbitrary order tensors and allows the short dimensions of the factor tensors to differ. We establish its connections with other classical tensor decompositions. Furthermore, implicit neural representation (INR) is employed to continuously represent the factor tensors in Multiple decomposition, enabling the method to generalize to non-grid data. We refer to this INR-based Multiple decomposition as Implicit Multiple Tensor Decomposition (IMTD). Then, the Proximal Alternating Least Squares (PALS) algorithm is utilized to solve the IMTD-based tensor reconstruction models. Since the objective function in IMTD-based models often lacks the Kurdyka-Lojasiewicz (KL) property, we establish a KL-free convergence analysis for the algorithm. Finally, extensive numerical experiments further validate the effectiveness of the proposed method.

Orthogonal finetuning (OFT) offers highly parameter-efficient adaptation while preventing catastrophic forgetting, but its high runtime and memory demands limit practical deployment. We identify the core computational bottleneck in OFT as its weight-centric implementation, which relies on costly matrix-matrix multiplications with cubic complexity. To overcome this, we propose OFTv2, an input-centric reformulation that instead uses matrix-vector multiplications (i.e., matrix-free computation), reducing the computational cost to quadratic. We further introduce the Cayley-Neumann parameterization, an efficient orthogonal parameterization that approximates the matrix inversion in Cayley transform via a truncated Neumann series. These modifications allow OFTv2 to achieve up to 10x faster training and 3x lower GPU memory usage without compromising performance. In addition, we extend OFTv2 to support finetuning quantized foundation models and show that it outperforms the popular QLoRA in training stability, efficiency, and memory usage.

We propose Orthogonal Monte Carlo Dropout, a mechanism that enforces strict orthogonality when combining sparse semantic vectors without extra time complexity. Low-Rank Adaptation (LoRA), a popular fine-tuning method for large models, typically trains a module to represent a specific concept such as an object or a style. When multiple LoRA modules are merged, for example to generate an object in a particular style, their outputs (semantic vectors) may interfere with each other. Our method guarantees that merged LoRA modules remain orthogonal and thus free from direct interference. However, empirical analysis reveals that such orthogonality does not lead to the semantic disentanglement highlighted in prior work on compositional adaptation. This finding suggests that inter-LoRA orthogonality alone may be insufficient for achieving true semantic compositionality, prompting a re-examination of its role in adapter merging.

Imposing orthogonality on the layers of neural networks is known to facilitate the learning by limiting the exploding/vanishing of the gradient; decorrelate the features; improve the robustness. This paper studies the theoretical properties of orthogonal convolutional layers.We establish necessary and sufficient conditions on the layer architecture guaranteeing the existence of an orthogonal convolutional transform. The conditions prove that orthogonal convolutional transforms exist for almost all architectures used in practice for 'circular' padding.We also exhibit limitations with 'valid' boundary conditions and 'same' boundary conditions with zero-padding.Recently, a regularization term imposing the orthogonality of convolutional layers has been proposed, and impressive empirical results have been obtained in different applications (Wang et al. 2020).The second motivation of the present paper is to specify the theory behind this.We make the link between this regularization term and orthogonality measures. In doing so, we show that this regularization strategy is stable with respect to numerical and optimization errors and that, in the presence of small errors and when the size of the signal/image is large, the convolutional layers remain close to isometric.The theoretical results are confirmed with experiments and the landscape of the regularization term is studied. Experiments on real data sets show that when orthogonality is used to enforce robustness, the parameter multiplying the regularization termcan be used to tune a tradeoff between accuracy and orthogonality, for the benefit of both accuracy and robustness.Altogether, the study guarantees that the regularization proposed in Wang et al. (2020) is an efficient, flexible and stable numerical strategy to learn orthogonal convolutional layers.

Deep Neural Networks (DNNs) have been a large driver and enabler for AI breakthroughs in recent years. These models have been getting larger in their attempt to become more accurate and tackle new upcoming use-cases, including AR/VR and intelligent assistants. However, the training process of such large models is a costly and time-consuming process, which typically yields a single model to fit all targets. To mitigate this, various techniques have been proposed in the literature, including pruning, sparsification or quantization of the model weights and updates. While able to achieve high compression rates, they often incur computational overheads or accuracy penalties. Alternatively, factorization methods have been leveraged to incorporate low-rank compression in the training process. Similarly, such techniques (e.g.,~SVD) frequently rely on the computationally expensive decomposition of layers and are potentially sub-optimal for non-linear models, such as DNNs. In this work, we take a further step in designing efficient low-rank models and propose Maestro, a framework for trainable low-rank layers. Instead of regularly applying a priori decompositions such as SVD, the low-rank structure is built into the training process through a generalized variant of Ordered Dropout. This method imposes an importance ordering via sampling on the decomposed DNN structure. Our theoretical analysis demonstrates that our method recovers the SVD decomposition of linear mapping on uniformly distributed data and PCA for linear autoencoders. We further apply our technique on DNNs and empirically illustrate that Maestro enables the extraction of lower footprint models that preserve model performance while allowing for graceful accuracy-latency tradeoff for the deployment to devices of different capabilities.

Orthogonal gradient updates have emerged as a promising direction in optimization for machine learning. However, traditional approaches such as SVD/QR decomposition incur prohibitive computational costs of O(n^3) and underperform compared to well-tuned SGD with momentum, since momentum is applied only after strict orthogonalization. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and producing semi-orthogonal matrices via Newton-Schulz iterations, reducing complexity to O(n^2). Nevertheless, quadratic costs remain a bottleneck. In this work, we study the semi-orthogonal properties of momentum-based updates and develop a method to bound momentum updates under a spectral-norm trust region, preserving directional information without requiring explicit semi-orthogonalization. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without constructing semi-orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. Our approach combines hyperbolic-cosine RMS scaling transformations with normalization, demonstrating both effectiveness and computational efficiency compared to Newton-Schulz methods. We further introduce a hybrid variant (Hybrid-AuON) that applies a single Newton-Schulz iteration. Experiments across vision and language benchmarks show that AuON and its hybrid variant achieve performance comparable to strong baselines such as AdamW and Muon. Code is available at: https://github.com/ryyzn9/AuON

The capabilities and adoption of deep neural networks (DNNs) grow at an exhilarating pace: Vision models accurately classify human actions in videos and identify cancerous tissue in medical scans as precisely than human experts; large language models answer wide-ranging questions, generate code, and write prose, becoming the topic of everyday dinner-table conversations. Even though their uses are exhilarating, the continually increasing model sizes and computational complexities have a dark side. The economic cost and negative environmental externalities of training and serving models is in evident disharmony with financial viability and climate action goals. Instead of pursuing yet another increase in predictive performance, this dissertation is dedicated to the improvement of neural network efficiency. Specifically, a core contribution addresses the efficiency aspects during online inference. Here, the concept of Continual Inference Networks (CINs) is proposed and explored across four publications. CINs extend prior state-of-the-art methods developed for offline processing of spatio-temporal data and reuse their pre-trained weights, improving their online processing efficiency by an order of magnitude. These advances are attained through a bottom-up computational reorganization and judicious architectural modifications. The benefit to online inference is demonstrated by reformulating several widely used network architectures into CINs, including 3D CNNs, ST-GCNs, and Transformer Encoders. An orthogonal contribution tackles the concurrent adaptation and computational acceleration of a large source model into multiple lightweight derived models. Drawing on fusible adapter networks and structured pruning, Structured Pruning Adapters achieve superior predictive accuracy under aggressive pruning using significantly fewer learned weights compared to fine-tuning with pruning.

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a spectral loss. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a 10times increase in performance speed.

Adapting large-scale pre-trained generative models in a parameter-efficient manner is gaining traction. Traditional methods like low rank adaptation achieve parameter efficiency by imposing constraints but may not be optimal for tasks requiring high representation capacity. We propose a novel spectrum-aware adaptation framework for generative models. Our method adjusts both singular values and their basis vectors of pretrained weights. Using the Kronecker product and efficient Stiefel optimizers, we achieve parameter-efficient adaptation of orthogonal matrices. We introduce Spectral Orthogonal Decomposition Adaptation (SODA), which balances computational efficiency and representation capacity. Extensive evaluations on text-to-image diffusion models demonstrate SODA's effectiveness, offering a spectrum-aware alternative to existing fine-tuning methods.

Fast linear transforms are ubiquitous in machine learning, including the discrete Fourier transform, discrete cosine transform, and other structured transformations such as convolutions. All of these transforms can be represented by dense matrix-vector multiplication, yet each has a specialized and highly efficient (subquadratic) algorithm. We ask to what extent hand-crafting these algorithms and implementations is necessary, what structural priors they encode, and how much knowledge is required to automatically learn a fast algorithm for a provided structured transform. Motivated by a characterization of fast matrix-vector multiplication as products of sparse matrices, we introduce a parameterization of divide-and-conquer methods that is capable of representing a large class of transforms. This generic formulation can automatically learn an efficient algorithm for many important transforms; for example, it recovers the O(N log N) Cooley-Tukey FFT algorithm to machine precision, for dimensions N up to 1024. Furthermore, our method can be incorporated as a lightweight replacement of generic matrices in machine learning pipelines to learn efficient and compressible transformations. On a standard task of compressing a single hidden-layer network, our method exceeds the classification accuracy of unconstrained matrices on CIFAR-10 by 3.9 points -- the first time a structured approach has done so -- with 4X faster inference speed and 40X fewer parameters.

Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in R^d. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a rm polylog(Delta)-approximation algorithm for embedding n-point ultrametrics into R^d with minimum distortion, where Delta denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

Low-rank decomposition has emerged as a vital tool for enhancing parameter efficiency in neural network architectures, gaining traction across diverse applications in machine learning. These techniques significantly lower the number of parameters, striking a balance between compactness and performance. However, a common challenge has been the compromise between parameter efficiency and the accuracy of the model, where reduced parameters often lead to diminished accuracy compared to their full-rank counterparts. In this work, we propose a novel theoretical framework that integrates a sinusoidal function within the low-rank decomposition process. This approach not only preserves the benefits of the parameter efficiency characteristic of low-rank methods but also increases the decomposition's rank, thereby enhancing model performance. Our method proves to be a plug in enhancement for existing low-rank models, as evidenced by its successful application in Vision Transformers (ViT), Large Language Models (LLMs), Neural Radiance Fields (NeRF) and 3D shape modelling.

Low Rank Decomposition of matrix - splitting a large matrix into a product of two smaller matrix offers a means for compression that reduces the parameters of a model without sparsification, and hence delivering more speedup on modern hardware. Moreover, unlike quantization, the compressed linear layers remain fully differentiable and all the parameters trainable, while being able to leverage the existing highly efficient kernels over floating point matrices. We study the potential to compress Large Language Models (LLMs) for monolingual Code generation via Low Rank Decomposition (LoRD) and observe that ranks for the linear layers in these models can be reduced by upto 39.58% with less than 1% increase in perplexity. We then use Low Rank Decomposition (LoRD) to compress StarCoder 16B to 13.2B parameter with no drop and to 12.3B with minimal drop in HumanEval Pass@1 score, in less than 10 minutes on a single A100. The compressed models speeds up inference by up to 22.35% with just a single line of change in code over huggingface's implementation with pytorch backend. Low Rank Decomposition (LoRD) models remain compatible with state of the art near-lossless quantization method such as SpQR, which allows leveraging further compression gains of quantization. Lastly, QLoRA over Low Rank Decomposition (LoRD) model further reduces memory requirements by as much as 21.2% over vanilla QLoRA while offering similar gains from parameter efficient fine tuning. Our work shows Low Rank Decomposition (LoRD) as a promising new paradigm for LLM compression.

A key step in reverse engineering neural networks is to decompose them into simpler parts that can be studied in relative isolation. Linear parameter decomposition -- a framework that has been proposed to resolve several issues with current decomposition methods -- decomposes neural network parameters into a sum of sparsely used vectors in parameter space. However, the current main method in this framework, Attribution-based Parameter Decomposition (APD), is impractical on account of its computational cost and sensitivity to hyperparameters. In this work, we introduce Stochastic Parameter Decomposition (SPD), a method that is more scalable and robust to hyperparameters than APD, which we demonstrate by decomposing models that are slightly larger and more complex than was possible to decompose with APD. We also show that SPD avoids other issues, such as shrinkage of the learned parameters, and better identifies ground truth mechanisms in toy models. By bridging causal mediation analysis and network decomposition methods, this demonstration opens up new research possibilities in mechanistic interpretability by removing barriers to scaling linear parameter decomposition methods to larger models. We release a library for running SPD and reproducing our experiments at https://github.com/goodfire-ai/spd.

Large foundation models are becoming ubiquitous, but training them from scratch is prohibitively expensive. Thus, efficiently adapting these powerful models to downstream tasks is increasingly important. In this paper, we study a principled finetuning paradigm -- Orthogonal Finetuning (OFT) -- for downstream task adaptation. Despite demonstrating good generalizability, OFT still uses a fairly large number of trainable parameters due to the high dimensionality of orthogonal matrices. To address this, we start by examining OFT from an information transmission perspective, and then identify a few key desiderata that enable better parameter-efficiency. Inspired by how the Cooley-Tukey fast Fourier transform algorithm enables efficient information transmission, we propose an efficient orthogonal parameterization using butterfly structures. We apply this parameterization to OFT, creating a novel parameter-efficient finetuning method, called Orthogonal Butterfly (BOFT). By subsuming OFT as a special case, BOFT introduces a generalized orthogonal finetuning framework. Finally, we conduct an extensive empirical study of adapting large vision transformers, large language models, and text-to-image diffusion models to various downstream tasks in vision and language.

We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.

In this paper, we propose a framework for solving a single-agent task by using multiple agents, each focusing on different aspects of the task. This approach has two main advantages: 1) it allows for training specialized agents on different parts of the task, and 2) it provides a new way to transfer knowledge, by transferring trained agents. Our framework generalizes the traditional hierarchical decomposition, in which, at any moment in time, a single agent has control until it has solved its particular subtask. We illustrate our framework with empirical experiments on two domains.

To adapt a well-trained large model to downstream tasks, we propose constraining learning within its original latent space by leveraging linear combinations of its basis vectors. This approach ensures stable training without compromising the model's capabilities. Traditionally, constructing orthonormal bases from a matrix requires a transfer matrix, which significantly increases storage and computational overhead for parameters and feature maps. In this paper, we introduce Absorb and Decompose for Q, K, V, and O matrices, enabling their orthogonalization without the need for transfer matrices. Furthermore, the Absorb-Decompose operation eliminates redundant vectors, reducing the encoder attention parameters of Whisper-large-v3 by 46.42% without requiring additional training. For parameter-efficient and stable fine-tuning, we orthonormalized Q, K, V, and O and fine-tuned only the singular values, allowing efficient adaptation while constraining changes to the original latent space. When fine-tuning LLaMA-2-7B on eight commonsense reasoning datasets, our method outperforms LoRA by 5.4% and DoRA by 4.4%.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

We study simplified bootstrap problems for probability distributions on the infinite line and the circle. We show that the rapid convergence of the bootstrap method for problems on the infinite line is related to the fact that the smallest eigenvalue of the positive matrices in the exact solution becomes exponentially small for large matrices, while the moments grow factorially. As a result, the positivity condition is very finely tuned. For problems on the circle we show instead that the entries of the positive matrix of Fourier modes of the distribution depend linearly on the initial data of the recursion, with factorially growing coefficients. By positivity, these matrix elements are bounded in absolute value by one, so the initial data must also be fine-tuned. Additionally, we find that we can largely bypass the semi-definite program (SDP) nature of the problem on a circle by recognizing that these Fourier modes must be asymptotically exponentially small. With a simple ansatz, which we call the shoestring bootstrap, we can efficiently identify an interior point of the set of allowed matrices with much higher precision than conventional SDP bounds permit. We apply this method to solving unitary matrix model integrals by numerically constructing the orthogonal polynomials associated with the circle distribution.

Factorizing a large matrix into small matrices is a popular strategy for model compression. Singular value decomposition (SVD) plays a vital role in this compression strategy, approximating a learned matrix with fewer parameters. However, SVD minimizes the squared error toward reconstructing the original matrix without gauging the importance of the parameters, potentially giving a larger reconstruction error for those who affect the task accuracy more. In other words, the optimization objective of SVD is not aligned with the trained model's task accuracy. We analyze this previously unexplored problem, make observations, and address it by introducing Fisher information to weigh the importance of parameters affecting the model prediction. This idea leads to our method: Fisher-Weighted SVD (FWSVD). Although the factorized matrices from our approach do not result in smaller reconstruction errors, we find that our resulting task accuracy is much closer to the original model's performance. We perform analysis with the transformer-based language models, showing our weighted SVD largely alleviates the mismatched optimization objectives and can maintain model performance with a higher compression rate. Our method can directly compress a task-specific model while achieving better performance than other compact model strategies requiring expensive model pre-training. Moreover, the evaluation of compressing an already compact model shows our method can further reduce 9% to 30% parameters with an insignificant impact on task accuracy.

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form h = g circ p where p : R^d rightarrow R is a degree k polynomial and g: R rightarrow R is a degree q polynomial. This function class generalizes the single-index model, which corresponds to k=1, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree k polynomials p, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target h up to vanishing test error in mathcal{O}(d^k) samples and polynomial time. This is a strict improvement over kernel methods, which require widetilde Theta(d^{kq}) samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of p being a quadratic. When p is indeed a quadratic, we achieve the information-theoretically optimal sample complexity mathcal{O}(d^2), which is an improvement over prior work~nichani2023provable requiring a sample size of widetildeTheta(d^4). Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature p with mathcal{O}(d^k) samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

With more open-source models available for diverse tasks, model merging has gained attention by combining models into one, reducing training, storage, and inference costs. Current research mainly focuses on model merging for full fine-tuning, overlooking the popular LoRA. However, our empirical analysis reveals that: a) existing merging methods designed for full fine-tuning perform poorly on LoRA; b) LoRA modules show much larger parameter magnitude variance than full fine-tuned weights; c) greater parameter magnitude variance correlates with worse merging performance. Considering that large magnitude variances cause deviations in the distribution of the merged parameters, resulting in information loss and performance degradation, we propose a Decoupled and Orthogonal merging approach(DO-Merging). By separating parameters into magnitude and direction components and merging them independently, we reduce the impact of magnitude differences on the directional alignment of the merged models, thereby preserving task information. Furthermore, we introduce a data-free, layer-wise gradient descent method with orthogonal constraints to mitigate interference during the merging of direction components. We provide theoretical guarantees for both the decoupling and orthogonal components. And we validate through extensive experiments across vision, language, and multi-modal domains that our proposed DO-Merging can achieve significantly higher performance than existing merging methods at a minimal cost. Notably, each component can be flexibly integrated with existing methods, offering near free-lunch improvements across tasks.

Retrieval systems rely on representations learned by increasingly powerful models. However, due to the high training cost and inconsistencies in learned representations, there is significant interest in facilitating communication between representations and ensuring compatibility across independently trained neural networks. In the literature, two primary approaches are commonly used to adapt different learned representations: affine transformations, which adapt well to specific distributions but can significantly alter the original representation, and orthogonal transformations, which preserve the original structure with strict geometric constraints but limit adaptability. A key challenge is adapting the latent spaces of updated models to align with those of previous models on downstream distributions while preserving the newly learned representation spaces. In this paper, we impose a relaxed orthogonality constraint, namely λ-Orthogonality regularization, while learning an affine transformation, to obtain distribution-specific adaptation while retaining the original learned representations. Extensive experiments across various architectures and datasets validate our approach, demonstrating that it preserves the model's zero-shot performance and ensures compatibility across model updates. Code available at: https://github.com/miccunifi/lambda_orthogonality.git{https://github.com/miccunifi/lambda\_orthogonality}.

While following different technical routes, both low-rank and orthogonal adaptation techniques can efficiently adapt large-scale pre-training models in specific tasks or domains based on a small piece of trainable parameters. In this study, we bridge the gap between these two techniques, proposing a simple but effective adaptation method based on Householder reflections. Given a pre-trained model, our method fine-tunes its layers by multiplying each frozen weight matrix with an orthogonal matrix constructed by a chain of learnable Householder reflections (HRs). This HR-based orthogonal fine-tuning is equivalent to an adaptive low-rank adaptation. Moreover, we show that the orthogonality of the reflection planes corresponding to the HRs impacts the model capacity and regularity. The analysis motivates us to regularize the orthogonality of the HRs, leading to different implementations of the proposed Householder reflection adaptation (HRA) method. Compared with state-of-the-art methods, HRA achieves superior performance with fewer learnable parameters when adapting large language models and conditional image generators. The code is available at https://github.com/DaShenZi721/HRA

Deep Click-Through Rate (CTR) prediction models play an important role in modern industrial recommendation scenarios. However, high memory overhead and computational costs limit their deployment in resource-constrained environments. Low-rank approximation is an effective method for computer vision and natural language processing models, but its application in compressing CTR prediction models has been less explored. Due to the limited memory and computing resources, compression of CTR prediction models often confronts three fundamental challenges, i.e., (1). How to reduce the model sizes to adapt to edge devices? (2). How to speed up CTR prediction model inference? (3). How to retain the capabilities of original models after compression? Previous low-rank compression research mostly uses tensor decomposition, which can achieve a high parameter compression ratio, but brings in AUC degradation and additional computing overhead. To address these challenges, we propose a unified low-rank decomposition framework for compressing CTR prediction models. We find that even with the most classic matrix decomposition SVD method, our framework can achieve better performance than the original model. To further improve the effectiveness of our framework, we locally compress the output features instead of compressing the model weights. Our unified low-rank compression framework can be applied to embedding tables and MLP layers in various CTR prediction models. Extensive experiments on two academic datasets and one real industrial benchmark demonstrate that, with 3-5x model size reduction, our compressed models can achieve both faster inference and higher AUC than the uncompressed original models. Our code is at https://github.com/yuhao318/Atomic_Feature_Mimicking.

Novel architectures have recently improved generative image synthesis leading to excellent visual quality in various tasks. Much of this success is due to the scalability of these architectures and hence caused by a dramatic increase in model complexity and in the computational resources invested in training these models. Our work questions the underlying paradigm of compressing large training data into ever growing parametric representations. We rather present an orthogonal, semi-parametric approach. We complement comparably small diffusion or autoregressive models with a separate image database and a retrieval strategy. During training we retrieve a set of nearest neighbors from this external database for each training instance and condition the generative model on these informative samples. While the retrieval approach is providing the (local) content, the model is focusing on learning the composition of scenes based on this content. As demonstrated by our experiments, simply swapping the database for one with different contents transfers a trained model post-hoc to a novel domain. The evaluation shows competitive performance on tasks which the generative model has not been trained on, such as class-conditional synthesis, zero-shot stylization or text-to-image synthesis without requiring paired text-image data. With negligible memory and computational overhead for the external database and retrieval we can significantly reduce the parameter count of the generative model and still outperform the state-of-the-art.

In this paper, we propose a highly parameter-efficient approach to scaling pre-trained language models (PLMs) to a deeper model depth. Unlike prior work that shares all parameters or uses extra blocks, we design a more capable parameter-sharing architecture based on matrix product operator (MPO). MPO decomposition can reorganize and factorize the information of a parameter matrix into two parts: the major part that contains the major information (central tensor) and the supplementary part that only has a small proportion of parameters (auxiliary tensors). Based on such a decomposition, our architecture shares the central tensor across all layers for reducing the model size and meanwhile keeps layer-specific auxiliary tensors (also using adapters) for enhancing the adaptation flexibility. To improve the model training, we further propose a stable initialization algorithm tailored for the MPO-based architecture. Extensive experiments have demonstrated the effectiveness of our proposed model in reducing the model size and achieving highly competitive performance.

The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among existing approaches, the Newton-Schulz iteration has emerged as a particularly effective solution, as it relies solely on matrix multiplications and thus achieves high computational efficiency on GPU hardware. Despite its efficiency, the method has inherent limitations - its coefficients are fixed and thus not optimized for a given matrix. In this paper we address this issue by proposing a Chebyshev-optimized version of Newton-Schulz (CANS). Based on the Chebyshev's alternance theorem, we theoretically derive optimal coefficients for the 3-rd order Newton-Schulz iteration and apply a Remez algorithm to compute optimal higher-degree polynomials. We leverage these polynomials to construct controlled approximate orthogonalization schemes, which is of interest in deep learning applications. Practically, we demonstrate the method's effectiveness in two key applications: orthogonalization in the Muon optimizer, and providing an efficient retraction alternative for Riemannian optimization on the Stiefel manifold.

With the increasingly powerful performances and enormous scales of Pretrained Language Models (PLMs), promoting parameter efficiency in fine-tuning has become a crucial need for effective and efficient adaptation to various downstream tasks. One representative line of fine-tuning methods is Orthogonal Fine-tuning (OFT), which rigorously preserves the angular distances within the parameter space to preserve the pretrained knowledge. Despite the empirical effectiveness, OFT still suffers low parameter efficiency at O(d^2) and limited capability of downstream adaptation. Inspired by Givens rotation, in this paper, we proposed quasi-Givens Orthogonal Fine-Tuning (qGOFT) to address the problems. We first use O(d) Givens rotations to accomplish arbitrary orthogonal transformation in SO(d) with provable equivalence, reducing parameter complexity from O(d^2) to O(d). Then we introduce flexible norm and relative angular adjustments under soft orthogonality regularization to enhance the adaptation capability of downstream semantic deviations. Extensive experiments on various tasks and PLMs validate the effectiveness of our methods.

The optimization of large language models (LLMs) remains a critical challenge, particularly as model scaling exacerbates sensitivity to algorithmic imprecision and training instability. Recent advances in optimizers have improved convergence efficiency through momentum orthogonalization, but suffer from two key robustness limitations: dimensional fragility in orthogonalization precision and vulnerability to outlier-induced noise. To address these robustness challenges, we introduce ROOT, a Robust Orthogonalized Optimizer that enhances training stability through dual robustness mechanisms. First, we develop a dimension-robust orthogonalization scheme using adaptive Newton iterations with fine-grained coefficients tailored to specific matrix sizes, ensuring consistent precision across diverse architectural configurations. Second, we introduce an optimization-robust framework via proximal optimization that suppresses outlier noise while preserving meaningful gradient directions. Extensive experiments demonstrate that ROOT achieves significantly improved robustness, with faster convergence and superior final performance compared to both Muon and Adam-based optimizers, particularly in noisy and non-convex scenarios. Our work establishes a new paradigm for developing robust and precise optimizers capable of handling the complexities of modern large-scale model training. The code will be available at https://github.com/huawei-noah/noah-research/tree/master/ROOT.

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a preconditioning procedure that accelerates Newton-Schulz convergence and reduces its computational cost. We evaluate its impact and show that the overhead of our preconditioning can be made negligible. Furthermore, the faster convergence it enables allows us to remove one iteration out of the usual five without degrading approximation quality. Our publicly available implementation achieves up to a 2.8x speedup in the Newton-Schulz approximation. We also show that this has a direct impact on end-to-end training runtime with 5-10% improvement in realistic training scenarios across two efficiency-focused tasks. On challenging language or vision tasks, we validate that our method maintains equal or superior model performance while improving runtime. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Our code is publicly available on github.

We study the computational limits of Low-Rank Adaptation (LoRA) update for finetuning transformer-based models using fine-grained complexity theory. Our key observation is that the existence of low-rank decompositions within the gradient computation of LoRA adaptation leads to possible algorithmic speedup. This allows us to (i) identify a phase transition behavior and (ii) prove the existence of nearly linear algorithms by controlling the LoRA update computation term by term, assuming the Strong Exponential Time Hypothesis (SETH). For the former, we identify a sharp transition in the efficiency of all possible rank-r LoRA update algorithms for transformers, based on specific norms resulting from the multiplications of the input sequence X, pretrained weights W^star, and adapter matrices alpha B A / r. Specifically, we derive a shared upper bound threshold for such norms and show that efficient (sub-quadratic) approximation algorithms of LoRA exist only below this threshold. For the latter, we prove the existence of nearly linear approximation algorithms for LoRA adaptation by utilizing the hierarchical low-rank structures of LoRA gradients and approximating the gradients with a series of chained low-rank approximations. To showcase our theory, we consider two practical scenarios: partial (e.g., only W_V and W_Q) and full adaptations (e.g., W_Q, W_V, and W_K) of weights in attention heads.

We give explicit low-rank bilinear non-commutative schemes for multiplying structured n times n matrices with 2 leq n leq 5, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic complexity. Our schemes are discovered over F_2 or F_3 and lifted to Z or Q. Using a flip graph search over tensor decompositions, we derive schemes for general, upper-triangular, lower-triangular, symmetric, and skew-symmetric inputs, as well as products of a structured matrix with its transpose. In particular, we obtain 4 times 4 rank-34 schemes: (i) multiplying a general matrix by its transpose using 10 recursive calls, improving the factor from 26/41 (0.634) to 8/13 (0.615); and (ii) multiplying an upper-triangular matrix by a general matrix using 12 recursive calls, improving the factor from 8/13 (0.615) to 22/37 (0.595). Additionally, using F_3 flip graphs, we discover schemes over Q that fundamentally require the inverse of 2, including a 2 times 2 symmetric-symmetric multiplication of rank 5 and a 3 times 3 skew-symmetric-general multiplication of rank 14 (improving upon AlphaTensor's 15).

In this paper, we study the ideas of composition and decomposition in the context of vector spaces, graphs and matroids. For vector spaces V_{AB}, treated as collection of row vectors, with specified column set Auplus B, we define V_{SP}lrarv V_{PQ}, Scap Q= emptyset, to be the collection of all vectors (f_S,f_Q) such that (f_S,f_P)in V_{SP}, (f_P,f_Q)in V_{PQ}. An analogous operation G_{SP}lrarg G_{PQ}equivd G_{PQ} can be defined in relation to graphs G_{SP}, G_{PQ}, on edge sets Suplus P, Puplus Q, respectively in terms of an overlapping subgraph G_P which gets deleted in the right side graph (see for instance the notion of k-sum oxley). For matroids we define the `linking' M_{SP}lrarm M_{PQ} equivd (M_{SP}vee M_{PQ})times (Suplus Q), denoting the contraction operation by 'times'. In each case, we examine how to minimize the size of the `overlap' set P, without affecting the right side entity. In the case of vector spaces, there is a polynomial time algorithm for achieving the minimum, which we present. Similar ideas work for graphs and for matroids under appropriate conditions. Next we consider the problem of decomposition. Here, in the case of vector spaces, the problem is to decompose V_{SQ} as V_{SP}lrarv V_{PQ}, with minimum size P. We give a polynomial time algorithm for this purpose. In the case of graphs and matroids we give a solution to this problem under certain restrictions.

Driven by the rapid growth of model parameters, parameter-efficient fine-tuning (PEFT) has become essential for adapting large models to diverse downstream tasks under constrained computational resources. Within this paradigm, orthogonal fine-tuning and its variants preserve semantic representations of pre-trained models, but struggle to achieve both expressiveness and efficiency in terms of parameter counts, memory, and computation. To overcome this limitation, we propose efficient Orthogonal Fine-Tuning with Principal Subspace adaptation (PSOFT), which confines orthogonal transformations to the principal subspace of pre-trained weights. Specifically, PSOFT constructs this subspace via matrix decomposition to enable compatible transformations with higher effective rank, establishes a theoretical condition that strictly maintains the geometry of this subspace for essential semantic preservation, and introduces efficient tunable vectors that gradually relax orthogonality during training to enhance adaptability. Extensive experiments on 35 NLP and CV tasks across four representative models demonstrate that PSOFT offers a practical and scalable solution to simultaneously achieve semantic preservation, expressiveness, and multi-dimensional efficiency in PEFT. The code is publicly available at https://github.com/fei407/PSOFT.

Recently proposed pyramidal models decompose the conventional forward and backward diffusion processes into multiple stages operating at varying resolutions. These models handle inputs with higher noise levels at lower resolutions, while less noisy inputs are processed at higher resolutions. This hierarchical approach significantly reduces the computational cost of inference in multi-step denoising models. However, existing open-source pyramidal video models have been trained from scratch and tend to underperform compared to state-of-the-art systems in terms of visual plausibility. In this work, we present a pipeline that converts a pretrained diffusion model into a pyramidal one through low-cost finetuning, achieving this transformation without degradation in quality of output videos. Furthermore, we investigate and compare various strategies for step distillation within pyramidal models, aiming to further enhance the inference efficiency. Our results are available at https://qualcomm-ai-research.github.io/PyramidalWan.

Initialization of parameters in deep neural networks has been shown to have a big impact on the performance of the networks (Mishkin & Matas, 2015). The initialization scheme devised by He et al, allowed convolution activations to carry a constrained mean which allowed deep networks to be trained effectively (He et al., 2015a). Orthogonal initializations and more generally orthogonal matrices in standard recurrent networks have been proved to eradicate the vanishing and exploding gradient problem (Pascanu et al., 2012). Majority of current initialization schemes do not take fully into account the intrinsic structure of the convolution operator. Using the duality of the Fourier transform and the convolution operator, Convolution Aware Initialization builds orthogonal filters in the Fourier space, and using the inverse Fourier transform represents them in the standard space. With Convolution Aware Initialization we noticed not only higher accuracy and lower loss, but faster convergence. We achieve new state of the art on the CIFAR10 dataset, and achieve close to state of the art on various other tasks.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.

We introduce Post-Optimization Model Edit (POME), a new algorithm that enhances the performance of fine-tuned large language models using only their pretrained and fine-tuned checkpoints, without requiring extra data or further optimization. The core idea is to apply a muon-style projection to ΔW, the difference between the fine-tuned and pretrained weights. This projection uses truncated singular value decomposition (SVD) to equalize the influence of dominant update directions and prune small singular values, which often represent noise. As a simple post-processing step, POME is completely decoupled from the training pipeline. It requires zero modifications and imposes no overhead, making it universally compatible with any optimizer or distributed framework. POME delivers consistent gains, boosting average performance by +2.5\% on GSM8K and +1.0\% on code generation. Its broad applicability -- from 7B foundation models to 72B RLHF-instructed models -- establishes it as a practical, zero-cost enhancement for any fine-tuning pipeline. Code is available at https://github.com/NUS-HPC-AI-Lab/POME.

The reflection superposition phenomenon is complex and widely distributed in the real world, which derives various simplified linear and nonlinear formulations of the problem. In this paper, based on the investigation of the weaknesses of existing models, we propose a more general form of the superposition model by introducing a learnable residue term, which can effectively capture residual information during decomposition, guiding the separated layers to be complete. In order to fully capitalize on its advantages, we further design the network structure elaborately, including a novel dual-stream interaction mechanism and a powerful decomposition network with a semantic pyramid encoder. Extensive experiments and ablation studies are conducted to verify our superiority over state-of-the-art approaches on multiple real-world benchmark datasets. Our code is publicly available at https://github.com/mingcv/DSRNet.

We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order power method (SS-HOPM) to a certain modified tensor, constructed from a matrix flattening of the original tensor; and using appropriate deflation steps. We obtain rigorous guarantees for SPM regarding convergence and global optima for input tensors of dimension d and order m of CP rank up to O(d^{lfloor m/2rfloor}), via results in classical algebraic geometry and optimization theory. As a by-product of our analysis we prove that SS-HOPM converges unconditionally, settling a conjecture in [Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM Journal on Matrix Analysis and Applications 32(4), 1095-1124 (2011)]. We present numerical experiments which demonstrate that SPM is efficient and robust to noise, being up to one order of magnitude faster than state-of-the-art CP decomposition algorithms in certain experiments. Furthermore, prior knowledge of the CP rank is not required by SPM.

A spanning tree T of graph G is a rho-approximate universal Steiner tree (UST) for root vertex r if, for any subset of vertices S containing r, the cost of the minimal subgraph of T connecting S is within a rho factor of the minimum cost tree connecting S in G. Busch et al. (FOCS 2012) showed that every graph admits 2^{O(log n)}-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions. We settle these open questions by giving polynomial-time algorithms for computing both O(log ^ 7 n)-approximate USTs and poly-logarithmic strong sparse partition hierarchies. For graphs with constant doubling dimension or constant pathwidth we improve this to O(log n)-approximate USTs and O(1) strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms. We reduce the existence of these objects to the previously studied cluster aggregation problem and what we call dangling nets.

Compressing a predefined deep neural network (DNN) into a compact sub-network with competitive performance is crucial in the efficient machine learning realm. This topic spans various techniques, from structured pruning to neural architecture search, encompassing both pruning and erasing operators perspectives. Despite advancements, existing methods suffers from complex, multi-stage processes that demand substantial engineering and domain knowledge, limiting their broader applications. We introduce the third-generation Only-Train-Once (OTOv3), which first automatically trains and compresses a general DNN through pruning and erasing operations, creating a compact and competitive sub-network without the need of fine-tuning. OTOv3 simplifies and automates the training and compression process, minimizes the engineering efforts required from users. It offers key technological advancements: (i) automatic search space construction for general DNNs based on dependency graph analysis; (ii) Dual Half-Space Projected Gradient (DHSPG) and its enhanced version with hierarchical search (H2SPG) to reliably solve (hierarchical) structured sparsity problems and ensure sub-network validity; and (iii) automated sub-network construction using solutions from DHSPG/H2SPG and dependency graphs. Our empirical results demonstrate the efficacy of OTOv3 across various benchmarks in structured pruning and neural architecture search. OTOv3 produces sub-networks that match or exceed the state-of-the-arts. The source code will be available at https://github.com/tianyic/only_train_once.

Diffusion models are emerging expressive generative models, in which a large number of time steps (inference steps) are required for a single image generation. To accelerate such tedious process, reducing steps uniformly is considered as an undisputed principle of diffusion models. We consider that such a uniform assumption is not the optimal solution in practice; i.e., we can find different optimal time steps for different models. Therefore, we propose to search the optimal time steps sequence and compressed model architecture in a unified framework to achieve effective image generation for diffusion models without any further training. Specifically, we first design a unified search space that consists of all possible time steps and various architectures. Then, a two stage evolutionary algorithm is introduced to find the optimal solution in the designed search space. To further accelerate the search process, we employ FID score between generated and real samples to estimate the performance of the sampled examples. As a result, the proposed method is (i).training-free, obtaining the optimal time steps and model architecture without any training process; (ii). orthogonal to most advanced diffusion samplers and can be integrated to gain better sample quality. (iii). generalized, where the searched time steps and architectures can be directly applied on different diffusion models with the same guidance scale. Experimental results show that our method achieves excellent performance by using only a few time steps, e.g. 17.86 FID score on ImageNet 64 times 64 with only four steps, compared to 138.66 with DDIM. The code is available at https://github.com/lilijiangg/AutoDiffusion.

Intrinsic decomposition is a fundamental mid-level vision problem that plays a crucial role in various inverse rendering and computational photography pipelines. Generating highly accurate intrinsic decompositions is an inherently under-constrained task that requires precisely estimating continuous-valued shading and albedo. In this work, we achieve high-resolution intrinsic decomposition by breaking the problem into two parts. First, we present a dense ordinal shading formulation using a shift- and scale-invariant loss in order to estimate ordinal shading cues without restricting the predictions to obey the intrinsic model. We then combine low- and high-resolution ordinal estimations using a second network to generate a shading estimate with both global coherency and local details. We encourage the model to learn an accurate decomposition by computing losses on the estimated shading as well as the albedo implied by the intrinsic model. We develop a straightforward method for generating dense pseudo ground truth using our model's predictions and multi-illumination data, enabling generalization to in-the-wild imagery. We present an exhaustive qualitative and quantitative analysis of our predicted intrinsic components against state-of-the-art methods. Finally, we demonstrate the real-world applicability of our estimations by performing otherwise difficult editing tasks such as recoloring and relighting.

Mixture of Experts (MoE) models scale capacity but often suffer from representation collapse and gradient instability. We propose Dynamic Subspace Composition (DSC), a framework that approximates context-dependent weights via a state-dependent, sparse expansion of a shared basis bank. Formally, DSC models the weight update as a residual trajectory within a Star- Shaped Domain, employing a Magnitude-Gated Simplex Interpolation to ensure continuity at the identity. Unlike standard Mixture-of-LoRAs, which incurs O(M rd) parameter complexity by retrieving independent rank-r matrices, DSC constructs a compositional rank-K approximation from decoupled unit-norm basis vectors. This reduces parameter complexity to O(M d) and memory traffic to O(Kd), while Frame-Theoretic regularization and spectral constraints provide rigorous worst-case bounds on the dynamic update. The code is available at https://github. com/VladimerKhasia/DSC

Hierarchical classification offers an approach to incorporate the concept of mistake severity by leveraging a structured, labeled hierarchy. However, decoding in such settings frequently relies on heuristic decision rules, which may not align with task-specific evaluation metrics. In this work, we propose a framework for the optimal decoding of an output probability distribution with respect to a target metric. We derive optimal decision rules for increasingly complex prediction settings, providing universal algorithms when candidates are limited to the set of nodes. In the most general case of predicting a subset of nodes, we focus on rules dedicated to the hierarchical hF_{beta} scores, tailored to hierarchical settings. To demonstrate the practical utility of our approach, we conduct extensive empirical evaluations, showcasing the superiority of our proposed optimal strategies, particularly in underdetermined scenarios. These results highlight the potential of our methods to enhance the performance and reliability of hierarchical classifiers in real-world applications. The code is available at https://github.com/RomanPlaud/hierarchical_decision_rules

The traditional Transformer model encounters challenges with variable-length input sequences, particularly in Hyperspectral Image Classification (HSIC), leading to efficiency and scalability concerns. To overcome this, we propose a pyramid-based hierarchical transformer (PyFormer). This innovative approach organizes input data hierarchically into segments, each representing distinct abstraction levels, thereby enhancing processing efficiency for lengthy sequences. At each level, a dedicated transformer module is applied, effectively capturing both local and global context. Spatial and spectral information flow within the hierarchy facilitates communication and abstraction propagation. Integration of outputs from different levels culminates in the final input representation. Experimental results underscore the superiority of the proposed method over traditional approaches. Additionally, the incorporation of disjoint samples augments robustness and reliability, thereby highlighting the potential of our approach in advancing HSIC. The source code is available at https://github.com/mahmad00/PyFormer.

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree tree where the weight of each node is the sum of the weights of its children. A treemap for tree is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of tree and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, polygonal partitions, that use convex polygons, have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio O(depth(tree)). However, depth(tree) still depends on the input tree. Hence we introduce a new type of treemaps, namely orthoconvex treemaps, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio. We also obtain several specialized results for single-level treemaps, that is, treemaps where the input tree has depth~1.

Most real-world graphs exhibit a hierarchical structure, which is often overlooked by existing graph generation methods. To address this limitation, we propose a novel graph generative network that captures the hierarchical nature of graphs and successively generates the graph sub-structures in a coarse-to-fine fashion. At each level of hierarchy, this model generates communities in parallel, followed by the prediction of cross-edges between communities using separate neural networks. This modular approach enables scalable graph generation for large and complex graphs. Moreover, we model the output distribution of edges in the hierarchical graph with a multinomial distribution and derive a recursive factorization for this distribution. This enables us to generate community graphs with integer-valued edge weights in an autoregressive manner. Empirical studies demonstrate the effectiveness and scalability of our proposed generative model, achieving state-of-the-art performance in terms of graph quality across various benchmark datasets. The code is available at https://github.com/Karami-m/HiGen_main.

Large Language Models (LLMs) have reshaped the landscape of artificial intelligence by demonstrating exceptional performance across various tasks. However, substantial computational requirements make their deployment challenging on devices with limited resources. Recently, compression methods using low-rank matrix techniques have shown promise, yet these often lead to degraded accuracy or introduce significant overhead in parameters and inference latency. This paper introduces Modular Decomposition (MoDeGPT), a novel structured compression framework that does not need recovery fine-tuning while resolving the above drawbacks. MoDeGPT partitions the Transformer block into modules comprised of matrix pairs and reduces the hidden dimensions via reconstructing the module-level outputs. MoDeGPT is developed based on a theoretical framework that utilizes three well-established matrix decomposition algorithms -- Nystr\"om approximation, CR decomposition, and SVD -- and applies them to our redefined transformer modules. Our comprehensive experiments show MoDeGPT, without backward propagation, matches or surpasses previous structured compression methods that rely on gradient information, and saves 98% of compute costs on compressing a 13B model. On Llama-2/3 and OPT models, MoDeGPT maintains 90-95% zero-shot performance with 25-30% compression rates. Moreover, the compression can be done on a single GPU within a few hours and increases the inference throughput by up to 46%.

Low-rank approximation is a fundamental technique in modern data analysis, widely utilized across various fields such as signal processing, machine learning, and natural language processing. Despite its ubiquity, the mechanics of low-rank approximation and its application in adaptation can sometimes be obscure, leaving practitioners and researchers with questions about its true capabilities and limitations. This paper seeks to clarify low-rank approximation and adaptation by offering a comprehensive guide that reveals their inner workings and explains their utility in a clear and accessible way. Our focus here is to develop a solid intuition for how low-rank approximation and adaptation operate, and why they are so effective. We begin with basic concepts and gradually build up to the mathematical underpinnings, ensuring that readers of all backgrounds can gain a deeper understanding of low-rank approximation and adaptation. We strive to strike a balance between informal explanations and rigorous mathematics, ensuring that both newcomers and experienced experts can benefit from this survey. Additionally, we introduce new low-rank decomposition and adaptation algorithms that have not yet been explored in the field, hoping that future researchers will investigate their potential applicability.

Sparse autoencoders (SAEs) are a technique for sparse decomposition of neural network activations into human-interpretable features. However, current SAEs suffer from feature absorption, where specialized features capture instances of general features creating representation holes, and feature composition, where independent features merge into composite representations. In this work, we introduce Orthogonal SAE (OrtSAE), a novel approach aimed to mitigate these issues by enforcing orthogonality between the learned features. By implementing a new training procedure that penalizes high pairwise cosine similarity between SAE features, OrtSAE promotes the development of disentangled features while scaling linearly with the SAE size, avoiding significant computational overhead. We train OrtSAE across different models and layers and compare it with other methods. We find that OrtSAE discovers 9% more distinct features, reduces feature absorption (by 65%) and composition (by 15%), improves performance on spurious correlation removal (+6%), and achieves on-par performance for other downstream tasks compared to traditional SAEs.

Image Retrieval is a fundamental task of obtaining images similar to the query one from a database. A common image retrieval practice is to firstly retrieve candidate images via similarity search using global image features and then re-rank the candidates by leveraging their local features. Previous learning-based studies mainly focus on either global or local image representation learning to tackle the retrieval task. In this paper, we abandon the two-stage paradigm and seek to design an effective single-stage solution by integrating local and global information inside images into compact image representations. Specifically, we propose a Deep Orthogonal Local and Global (DOLG) information fusion framework for end-to-end image retrieval. It attentively extracts representative local information with multi-atrous convolutions and self-attention at first. Components orthogonal to the global image representation are then extracted from the local information. At last, the orthogonal components are concatenated with the global representation as a complementary, and then aggregation is performed to generate the final representation. The whole framework is end-to-end differentiable and can be trained with image-level labels. Extensive experimental results validate the effectiveness of our solution and show that our model achieves state-of-the-art image retrieval performances on Revisited Oxford and Paris datasets.

Efficiently capturing the long-range patterns in sequential data sources salient to a given task -- such as classification and generative modeling -- poses a fundamental challenge. Popular approaches in the space tradeoff between the memory burden of brute-force enumeration and comparison, as in transformers, the computational burden of complicated sequential dependencies, as in recurrent neural networks, or the parameter burden of convolutional networks with many or large filters. We instead take inspiration from wavelet-based multiresolution analysis to define a new building block for sequence modeling, which we call a MultiresLayer. The key component of our model is the multiresolution convolution, capturing multiscale trends in the input sequence. Our MultiresConv can be implemented with shared filters across a dilated causal convolution tree. Thus it garners the computational advantages of convolutional networks and the principled theoretical motivation of wavelet decompositions. Our MultiresLayer is straightforward to implement, requires significantly fewer parameters, and maintains at most a O(Nlog N) memory footprint for a length N sequence. Yet, by stacking such layers, our model yields state-of-the-art performance on a number of sequence classification and autoregressive density estimation tasks using CIFAR-10, ListOps, and PTB-XL datasets.

Current learned image compression models typically exhibit high complexity, which demands significant computational resources. To overcome these challenges, we propose an innovative approach that employs hierarchical feature extraction transforms to significantly reduce complexity while preserving bit rate reduction efficiency. Our novel architecture achieves this by using fewer channels for high spatial resolution inputs/feature maps. On the other hand, feature maps with a large number of channels have reduced spatial dimensions, thereby cutting down on computational load without sacrificing performance. This strategy effectively reduces the forward pass complexity from \(1256 \, kMAC/Pixel\) to just \(270 \, kMAC/Pixel\). As a result, the reduced complexity model can open the way for learned image compression models to operate efficiently across various devices and pave the way for the development of new architectures in image compression technology.

Large text-to-image diffusion models have impressive capabilities in generating photorealistic images from text prompts. How to effectively guide or control these powerful models to perform different downstream tasks becomes an important open problem. To tackle this challenge, we introduce a principled finetuning method -- Orthogonal Finetuning (OFT), for adapting text-to-image diffusion models to downstream tasks. Unlike existing methods, OFT can provably preserve hyperspherical energy which characterizes the pairwise neuron relationship on the unit hypersphere. We find that this property is crucial for preserving the semantic generation ability of text-to-image diffusion models. To improve finetuning stability, we further propose Constrained Orthogonal Finetuning (COFT) which imposes an additional radius constraint to the hypersphere. Specifically, we consider two important finetuning text-to-image tasks: subject-driven generation where the goal is to generate subject-specific images given a few images of a subject and a text prompt, and controllable generation where the goal is to enable the model to take in additional control signals. We empirically show that our OFT framework outperforms existing methods in generation quality and convergence speed.

The industrial application motivating this work is the fatigue computation of aircraft engines' high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and can reach values relatively close to the creep temperature: important nonlinear effects occur and the solution strongly depends on the used thermal loading. We consider a nonlinear reduced order model able to compute, in the exploitation phase, the behavior of the blade for a new temperature field loading. The sensitivity of the solution to the temperature makes {the classical unenriched proper orthogonal decomposition method} fail. In this work, we propose a new error indicator, quantifying the error made by the reduced order model in computational complexity independent of the size of the high-fidelity reference model. In our framework, when the {error indicator} becomes larger than a given tolerance, the reduced order model is updated using one time step solution of the high-fidelity reference model. The approach is illustrated on a series of academic test cases and applied on a setting of industrial complexity involving 5 million degrees of freedom, where the whole procedure is computed in parallel with distributed memory.

Hierarchical clustering of networks consists in finding a tree of communities, such that lower levels of the hierarchy reveal finer-grained community structures. There are two main classes of algorithms tackling this problem. Divisive (top-down) algorithms recursively partition the nodes into two communities, until a stopping rule indicates that no further split is needed. In contrast, agglomerative (bottom-up) algorithms first identify the smallest community structure and then repeatedly merge the communities using a linkage method. In this article, we establish theoretical guarantees for the recovery of the hierarchical tree and community structure of a Hierarchical Stochastic Block Model by a bottom-up algorithm. We also establish that this bottom-up algorithm attains the information-theoretic threshold for exact recovery at intermediate levels of the hierarchy. Notably, these recovery conditions are less restrictive compared to those existing for top-down algorithms. This shows that bottom-up algorithms extend the feasible region for achieving exact recovery at intermediate levels. Numerical experiments on both synthetic and real data sets confirm the superiority of bottom-up algorithms over top-down algorithms. We also observe that top-down algorithms can produce dendrograms with inversions. These findings contribute to a better understanding of hierarchical clustering techniques and their applications in network analysis.

Representation learning and exploration are among the key challenges for any deep reinforcement learning agent. In this work, we provide a singular value decomposition based method that can be used to obtain representations that preserve the underlying transition structure in the domain. Perhaps interestingly, we show that these representations also capture the relative frequency of state visitations, thereby providing an estimate for pseudo-counts for free. To scale this decomposition method to large-scale domains, we provide an algorithm that never requires building the transition matrix, can make use of deep networks, and also permits mini-batch training. Further, we draw inspiration from predictive state representations and extend our decomposition method to partially observable environments. With experiments on multi-task settings with partially observable domains, we show that the proposed method can not only learn useful representation on DM-Lab-30 environments (that have inputs involving language instructions, pixel images, and rewards, among others) but it can also be effective at hard exploration tasks in DM-Hard-8 environments.

Customization techniques for text-to-image models have paved the way for a wide range of previously unattainable applications, enabling the generation of specific concepts across diverse contexts and styles. While existing methods facilitate high-fidelity customization for individual concepts or a limited, pre-defined set of them, they fall short of achieving scalability, where a single model can seamlessly render countless concepts. In this paper, we address a new problem called Modular Customization, with the goal of efficiently merging customized models that were fine-tuned independently for individual concepts. This allows the merged model to jointly synthesize concepts in one image without compromising fidelity or incurring any additional computational costs. To address this problem, we introduce Orthogonal Adaptation, a method designed to encourage the customized models, which do not have access to each other during fine-tuning, to have orthogonal residual weights. This ensures that during inference time, the customized models can be summed with minimal interference. Our proposed method is both simple and versatile, applicable to nearly all optimizable weights in the model architecture. Through an extensive set of quantitative and qualitative evaluations, our method consistently outperforms relevant baselines in terms of efficiency and identity preservation, demonstrating a significant leap toward scalable customization of diffusion models.

LLMs have achieved remarkable breakthroughs in reasoning, insights, and tool use, but chaining these abilities into extended processes at the scale of those routinely executed by humans, organizations, and societies has remained out of reach. The models have a persistent error rate that prevents scale-up: for instance, recent experiments in the Towers of Hanoi benchmark domain showed that the process inevitably becomes derailed after at most a few hundred steps. Thus, although LLM research is often still benchmarked on tasks with relatively few dependent logical steps, there is increasing attention on the ability (or inability) of LLMs to perform long range tasks. This paper describes MAKER, the first system that successfully solves a task with over one million LLM steps with zero errors, and, in principle, scales far beyond this level. The approach relies on an extreme decomposition of a task into subtasks, each of which can be tackled by focused microagents. The high level of modularity resulting from the decomposition allows error correction to be applied at each step through an efficient multi-agent voting scheme. This combination of extreme decomposition and error correction makes scaling possible. Thus, the results suggest that instead of relying on continual improvement of current LLMs, massively decomposed agentic processes (MDAPs) may provide a way to efficiently solve problems at the level of organizations and societies.

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central approach to manifold learning, and hyperbolic geometry. Specifically, using diffusion geometry, we build multi-scale densities on the data, aimed to reveal their hierarchical structure, and then embed them into a product of hyperbolic spaces. We show theoretically that our embedding and distance recover the underlying hierarchical structure. In addition, we demonstrate the efficacy of the proposed method and its advantages compared to existing methods on graph embedding benchmarks and hierarchical datasets.

We present a new approach for the approximate K-nearest neighbor search based on navigable small world graphs with controllable hierarchy (Hierarchical NSW, HNSW). The proposed solution is fully graph-based, without any need for additional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. Hierarchical NSW incrementally builds a multi-layer structure consisting from hierarchical set of proximity graphs (layers) for nested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an exponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Small World (NSW) structures while additionally having the links separated by their characteristic distance scales. Starting search from the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a logarithmic complexity scaling. Additional employment of a heuristic for selecting proximity graph neighbors significantly increases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the proposed general metric space search index is able to strongly outperform previous opensource state-of-the-art vector-only approaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation.

Fine-tuning large language models (LLMs) with high parameter efficiency for downstream tasks has become a new paradigm. Low-Rank Adaptation (LoRA) significantly reduces the number of trainable parameters for fine-tuning. Although it has demonstrated commendable performance, updating parameters within a single scale may not be the optimal choice for complex downstream tasks.In this paper, we extend the LoRA to multiple scales, dubbed as LoRA^2. We first combine orthogonal projection theory to train a set of LoRAs in two mutually orthogonal planes. Then, we improve the importance score algorithm, which reduce parameter sensitivity score calculations by approximately 98.5\%. By pruning singular values with lower importance scores, thereby enhancing adaptability to various downstream tasks. Extensive experiments are conducted on two widely used pre-trained models to validate the effectiveness of LoRA^2. Results show that it significantly reduces the number of trainable parameters to just 0.72\% compared to full fine-tuning, while still delivering highly impressive performance. Even when the parameters are further reduced to 0.17M, it still achieves comparable results to the baseline with 8 times more parameters. Our code is available here: https://anonymous.4open.science/r/LoRA-2-5B4C

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Mixture of Experts (MoE) LLMs face significant obstacles due to their massive parameter scale, which imposes memory, storage, and deployment challenges. Although recent expert merging methods promise greater efficiency by consolidating multiple experts, they are fundamentally hindered by parameter conflicts arising from expert specialization. In this paper, we present Sub-MoE, a novel MoE compression framework via Subspace Expert Merging. Our key insight is to perform joint Singular Value Decomposition (SVD) on concatenated expert weights, reducing conflicting parameters by extracting shared U-matrices while enabling effective merging of the expert-specific V components. Specifically, Sub-MoE consists of two innovative phases: (1) Adaptive Expert Clustering, which groups functionally coherent experts via K-means clustering based on cosine similarity of expert outputs; and (2) Subspace Expert Merging, which first enforces Experts Union Decomposition to derive the shared U-matrix across experts in the same group, then pursues frequency-based merging for individual V-matrices, and finalizes expert reconstruction using the merged V-matrix. In this way, we align and fuse experts in a shared subspace, and can be extended with intra-expert compression for further inference optimization. Extensive experiments on Mixtral, DeepSeek, and Qwen-1.5|3 MoE LLMs demonstrate that our Sub-MoE significantly outperforms existing expert pruning and merging methods. Notably, our Sub-MoE maintains 96\%|86\% of original performance with 25\%|50\% expert reduction on Mixtral-8x7B in zero-shot benchmarks. Code will be released at https://github.com/lliai/MoERazor.

This paper presents algorithms for hierarchical, agglomerative clustering which perform most efficiently in the general-purpose setup that is given in modern standard software. Requirements are: (1) the input data is given by pairwise dissimilarities between data points, but extensions to vector data are also discussed (2) the output is a "stepwise dendrogram", a data structure which is shared by all implementations in current standard software. We present algorithms (old and new) which perform clustering in this setting efficiently, both in an asymptotic worst-case analysis and from a practical point of view. The main contributions of this paper are: (1) We present a new algorithm which is suitable for any distance update scheme and performs significantly better than the existing algorithms. (2) We prove the correctness of two algorithms by Rohlf and Murtagh, which is necessary in each case for different reasons. (3) We give well-founded recommendations for the best current algorithms for the various agglomerative clustering schemes.

This paper introduces a novel hierarchical autoencoder that maps 3D models into a highly compressed latent space. The hierarchical autoencoder is specifically designed to tackle the challenges arising from large-scale datasets and generative modeling using diffusion. Different from previous approaches that only work on a regular image or volume grid, our hierarchical autoencoder operates on unordered sets of vectors. Each level of the autoencoder controls different geometric levels of detail. We show that the model can be used to represent a wide range of 3D models while faithfully representing high-resolution geometry details. The training of the new architecture takes 0.70x time and 0.58x memory compared to the baseline. We also explore how the new representation can be used for generative modeling. Specifically, we propose a cascaded diffusion framework where each stage is conditioned on the previous stage. Our design extends existing cascaded designs for image and volume grids to vector sets.

Task arithmetic provides an efficient, training-free way to edit pre-trained models, yet lacks a fundamental theoretical explanation for its success. The existing concept of ``weight disentanglement" describes the ideal outcome of non-interfering task composition but does not reveal its underlying cause. Crucially, what intrinsic properties of the pre-trained model (θ_0) or the task vectors (τ_t) enable this disentanglement remains underexplored. In this paper, we introduce Task-Feature Specialization (TFS), a model's ability to allocate distinct internal features to different tasks, as the fundamental principle. We first prove that TFS is a sufficient condition for weight disentanglement. More importantly, we find that TFS also gives rise to an observable geometric consequence: weight vector orthogonality. This positions TFS as the common cause for both the desired functional outcome (disentanglement) and a measurable geometric property (orthogonality). This relationship provides the key insight for our method: since the abstract TFS property is intractable to enforce directly, we can instead promote weight disentanglement by shaping its concrete geometric consequence, orthogonality. Therefore, we propose OrthoReg, a simple and effective regularization method that actively enforces an internal orthogonal structure on weight updates (ΔW) that constitute τ_t during fine-tuning. And we theoretically prove that OrthoReg promotes disentanglement. Extensive experiments demonstrate that OrthoReg consistently and significantly enhances the performance of various task arithmetic methods. Code is available at https://github.com/RL-MIND/OrthoReg{https://github.com/RL-MIND/OrthoReg}.

Singular Value Decomposition (SVD) has become an important technique for reducing the computational burden of Vision Language Models (VLMs), which play a central role in tasks such as image captioning and visual question answering. Although multiple prior works have proposed efficient SVD variants to enable low-rank operations, we find that in practice it remains difficult to achieve substantial latency reduction during model execution. To address this limitation, we introduce a new computational pattern and apply SVD at a finer granularity, enabling real and measurable improvements in execution latency. Furthermore, recognizing that weight elements differ in their relative importance, we adaptively allocate relative importance to each element during SVD process to better preserve accuracy, then extend this framework with quantization applied to both weights and activations, resulting in a highly efficient VLM. Collectively, we introduce~Weighted SVD (WSVD), which outperforms other approaches by achieving over 1.8times decoding speedup while preserving accuracy. We open source our code at: https://github.com/SAI-Lab-NYU/WSVD{https://github.com/SAI-Lab-NYU/WSVD

Recent advances to combine structured regression models and deep neural networks for better interpretability, more expressiveness, and statistically valid uncertainty quantification demonstrate the versatility of semi-structured neural networks (SSNs). We show that techniques to properly identify the contributions of the different model components in SSNs, however, lead to suboptimal network estimation, slower convergence, and degenerated or erroneous predictions. In order to solve these problems while preserving favorable model properties, we propose a non-invasive post-hoc orthogonalization (PHO) that guarantees identifiability of model components and provides better estimation and prediction quality. Our theoretical findings are supported by numerical experiments, a benchmark comparison as well as a real-world application to COVID-19 infections.

Many practical applications, such as recommender systems and learning to rank, involve solving multiple similar tasks. One example is learning of recommendation policies for users with similar movie preferences, where the users may still rank the individual movies slightly differently. Such tasks can be organized in a hierarchy, where similar tasks are related through a shared structure. In this work, we formulate this problem as a contextual off-policy optimization in a hierarchical graphical model from logged bandit feedback. To solve the problem, we propose a hierarchical off-policy optimization algorithm (HierOPO), which estimates the parameters of the hierarchical model and then acts pessimistically with respect to them. We instantiate HierOPO in linear Gaussian models, for which we also provide an efficient implementation and analysis. We prove per-task bounds on the suboptimality of the learned policies, which show a clear improvement over not using the hierarchical model. We also evaluate the policies empirically. Our theoretical and empirical results show a clear advantage of using the hierarchy over solving each task independently.

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

Low-Rank Adaptation (LoRA) offers a parameter-efficient paradigm for tuning large models. While recent spectral initialization methods improve convergence and performance over the naive "Noise & Zeros" scheme, their extra computational and storage overhead undermines efficiency. In this paper, we establish update magnitude as the fundamental driver of LoRA performance and propose LoRAM, a magnitude-driven "Basis & Basis" initialization scheme that matches spectral methods without their inefficiencies. Our key contributions are threefold: (i) Magnitude of weight updates determines convergence. We prove low-rank structures intrinsically bound update magnitudes, unifying hyperparameter tuning in learning rate, scaling factor, and initialization as mechanisms to optimize magnitude regulation. (ii) Spectral initialization succeeds via magnitude amplification. We demystify that the presumed knowledge-driven benefit of the spectral component essentially arises from the boost in the weight update magnitude. (iii) A novel and compact initialization strategy, LoRAM, scales deterministic orthogonal bases using pretrained weight magnitudes to simulate spectral gains. Extensive experiments show that LoRAM serves as a strong baseline, retaining the full efficiency of LoRA while matching or outperforming spectral initialization across benchmarks.

With the growth of neural network size, model compression has attracted increasing interest in recent research. As one of the most common techniques, pruning has been studied for a long time. By exploiting the structured sparsity of the neural network, existing methods can prune neurons instead of individual weights. However, in most existing pruning methods, surviving neurons are randomly connected in the neural network without any structure, and the non-zero weights within each neuron are also randomly distributed. Such irregular sparse structure can cause very high control overhead and irregular memory access for the hardware and even increase the neural network computational complexity. In this paper, we propose a three-layer hierarchical prior to promote a more regular sparse structure during pruning. The proposed three-layer hierarchical prior can achieve per-neuron weight-level structured sparsity and neuron-level structured sparsity. We derive an efficient Turbo-variational Bayesian inferencing (Turbo-VBI) algorithm to solve the resulting model compression problem with the proposed prior. The proposed Turbo-VBI algorithm has low complexity and can support more general priors than existing model compression algorithms. Simulation results show that our proposed algorithm can promote a more regular structure in the pruned neural networks while achieving even better performance in terms of compression rate and inferencing accuracy compared with the baselines.

KV cache has become a de facto technique for the inference of large language models (LLMs), where tensors of shape (layer number, head number, sequence length, feature dimension) are introduced to cache historical information for self-attention. As the size of the model and data grows, the KV cache can quickly become a bottleneck within the system in both storage and memory transfer. To address this, prior studies usually focus on the first three axes of the cache tensors for compression. This paper supplements them, focusing on the feature dimension axis, by utilizing low-rank projection matrices to transform the cache features into spaces with reduced dimensions. We begin by investigating the canonical orthogonal projection method for data compression through principal component analysis (PCA). We observe the issue with PCA projection where significant performance degradation is observed at low compression rates. To bridge the gap, we propose to directly tune the orthogonal projection matrices with a distillation objective using an elaborate Matryoshka training strategy. After training, we adaptively search for the optimal compression rates for various layers and heads given varying compression budgets. Compared to previous works, our method can easily embrace pre-trained LLMs and hold a smooth tradeoff between performance and compression rate. We empirically witness the high data efficiency of our training procedure and find that our method can sustain over 90% performance with an average KV cache compression rate of 60% (and up to 75% in certain extreme scenarios) for popular LLMs like LLaMA2-7B-base and Mistral-7B-v0.3-base.

Neural operators, as an efficient surrogate model for learning the solutions of PDEs, have received extensive attention in the field of scientific machine learning. Among them, attention-based neural operators have become one of the mainstreams in related research. However, existing approaches overfit the limited training data due to the considerable number of parameters in the attention mechanism. To address this, we develop an orthogonal attention based on the eigendecomposition of the kernel integral operator and the neural approximation of eigenfunctions. The orthogonalization naturally poses a proper regularization effect on the resulting neural operator, which aids in resisting overfitting and boosting generalization. Experiments on six standard neural operator benchmark datasets comprising both regular and irregular geometries show that our method can outperform competing baselines with decent margins.

Recently, using diffusion models for zero-shot image restoration (IR) has become a new hot paradigm. This type of method only needs to use the pre-trained off-the-shelf diffusion models, without any finetuning, and can directly handle various IR tasks. The upper limit of the restoration performance depends on the pre-trained diffusion models, which are in rapid evolution. However, current methods only discuss how to deal with fixed-size images, but dealing with images of arbitrary sizes is very important for practical applications. This paper focuses on how to use those diffusion-based zero-shot IR methods to deal with any size while maintaining the excellent characteristics of zero-shot. A simple way to solve arbitrary size is to divide it into fixed-size patches and solve each patch independently. But this may yield significant artifacts since it neither considers the global semantics of all patches nor the local information of adjacent patches. Inspired by the Range-Null space Decomposition, we propose the Mask-Shift Restoration to address local incoherence and propose the Hierarchical Restoration to alleviate out-of-domain issues. Our simple, parameter-free approaches can be used not only for image restoration but also for image generation of unlimited sizes, with the potential to be a general tool for diffusion models. Code: https://github.com/wyhuai/DDNM/tree/main/hq_demo

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

Matrix-based optimizers have attracted growing interest for improving LLM training efficiency, with significant progress centered on orthogonalization/whitening based methods. While yielding substantial performance gains, a fundamental question arises: can we develop new paradigms beyond orthogonalization, pushing the efficiency frontier further? We present Adaptively Rotated Optimization (ARO, a new matrix optimization framework that treats gradient rotation as a first class design principle. ARO accelerates LLM training by performing normed steepest descent in a rotated coordinate system, where the rotation is determined by a novel norm-informed policy. This perspective yields update rules that go beyond existing orthogonalization and whitening optimizers, improving sample efficiency in practice. To make comparisons reliable, we propose a rigorously controlled benchmarking protocol that reduces confounding and bias. Under this protocol, ARO consistently outperforms AdamW (by 1.3 sim1.35times) and orthogonalization methods (by 1.1sim1.15times) in LLM pretraining at up to 8B activated parameters, and up to 8times overtrain budget, without evidence of diminishing returns. Finally, we discuss how ARO can be reformulated as a symmetry-aware optimizer grounded in rotational symmetries of residual streams, motivating advanced designs that enable computationally efficient exploitation of cross-layer/cross module couplings.

Proposing an effective and flexible matrix to represent a graph is a fundamental challenge that has been explored from multiple perspectives, e.g., filtering in Graph Fourier Transforms. In this work, we develop a novel and general framework which unifies many existing GNN models from the view of parameterized decomposition and filtering, and show how it helps to enhance the flexibility of GNNs while alleviating the smoothness and amplification issues of existing models. Essentially, we show that the extensively studied spectral graph convolutions with learnable polynomial filters are constrained variants of this formulation, and releasing these constraints enables our model to express the desired decomposition and filtering simultaneously. Based on this generalized framework, we develop models that are simple in implementation but achieve significant improvements and computational efficiency on a variety of graph learning tasks. Code is available at https://github.com/qslim/PDF.

Multivariate time series is prevalent in many scientific and industrial domains. Modeling multivariate signals is challenging due to their long-range temporal dependencies and intricate interactions--both direct and indirect. To confront these complexities, we introduce a method of representing multivariate signals as nodes in a graph with edges indicating interdependency between them. Specifically, we leverage graph neural networks (GNN) and attention mechanisms to efficiently learn the underlying relationships within the time series data. Moreover, we suggest employing hierarchical signal decompositions running over the graphs to capture multiple spatial dependencies. The effectiveness of our proposed model is evaluated across various real-world benchmark datasets designed for long-term forecasting tasks. The results consistently showcase the superiority of our model, achieving an average 23\% reduction in mean squared error (MSE) compared to existing models.

Approximations to Gaussian processes based on inducing variables, combined with variational inference techniques, enable state-of-the-art sparse approaches to infer GPs at scale through mini batch-based learning. In this work, we address one limitation of sparse GPs, which is due to the challenge in dealing with a large number of inducing variables without imposing a special structure on the inducing inputs. In particular, we introduce a novel hierarchical prior, which imposes sparsity on the set of inducing variables. We treat our model variationally, and we experimentally show considerable computational gains compared to standard sparse GPs when sparsity on the inducing variables is realized considering the nearest inducing inputs of a random mini-batch of the data. We perform an extensive experimental validation that demonstrates the effectiveness of our approach compared to the state-of-the-art. Our approach enables the possibility to use sparse GPs using a large number of inducing points without incurring a prohibitive computational cost.

Mechanistic interpretability seeks to reverse-engineer neural network computations into human-understandable algorithms, yet extracting sparse computational circuits from billion-parameter language models remains challenging due to exponential search complexity and pervasive polysemanticity. The proposed Hierarchical Attribution Graph Decomposition (HAGD) framework reduces circuit discovery complexity from O(2^n) exhaustive enumeration to O(n^2 log n) through multi-resolution abstraction hierarchies and differentiable circuit search. The methodology integrates cross-layer transcoders for monosemantic feature extraction, graph neural network meta-learning for topology prediction, and causal intervention protocols for validation. Empirical evaluation spans GPT-2 variants, Llama-7B through Llama-70B, and Pythia suite models across algorithmic tasks and natural language benchmarks. On modular arithmetic tasks, the framework achieves up to 91% behavioral preservation (pm2.3\% across runs) while maintaining interpretable subgraph sizes. Cross-architecture transfer experiments suggest that discovered circuits exhibit moderate structural similarity (averaging 67%) across model families, indicating potential shared computational patterns. These results provide preliminary foundations for interpretability at larger model scales while identifying significant limitations in current attribution methodologies that require future advances.

Is the standard weight decay in AdamW truly optimal? Although AdamW decouples weight decay from adaptive gradient scaling, a fundamental conflict remains: the Radial Tug-of-War. In deep learning, gradients tend to increase parameter norms to expand effective capacity while steering directions to learn features, whereas weight decay indiscriminately suppresses norm growth. This push--pull interaction induces radial oscillations, injecting noise into Adam's second-moment estimates and potentially degrading delicate tangential feature learning. We argue that magnitude and direction play distinct roles and should be decoupled in optimizer dynamics. We propose Orthogonal Dynamics Decoupling and instantiate it as AdamO: an SGD-style update handles the one-dimensional norm control, while Adam's adaptive preconditioning is confined to the tangential subspace. AdamO further incorporates curvature-adaptive radial step sizing and architecture-aware rules and projections for scale-invariant layers and low-dimensional parameters. Experiments on vision and language tasks show that AdamO improves generalization and stability over AdamW without introducing additional complex constraints.

Given a massive graph, how can we exploit its hierarchical structure for concisely but exactly summarizing the graph? By exploiting the structure, can we achieve better compression rates than state-of-the-art graph summarization methods? The explosive proliferation of the Web has accelerated the emergence of large graphs, such as online social networks and hyperlink networks. Consequently, graph compression has become increasingly important to process such large graphs without expensive I/O over the network or to disk. Among a number of approaches, graph summarization, which in essence combines similar nodes into a supernode and describe their connectivity concisely, protrudes with several advantages. However, we note that it fails to exploit pervasive hierarchical structures of real-world graphs as its underlying representation model enforces supernodes to be disjoint. In this work, we propose the hierarchical graph summarization model, which is an expressive graph representation model that includes the previous one proposed by Navlakha et al. as a special case. The new model represents an unweighted graph using positive and negative edges between hierarchical supernodes, each of which can contain others. Then, we propose Slugger, a scalable heuristic for concisely and exactly representing a given graph under our new model. Slugger greedily merges nodes into supernodes while maintaining and exploiting their hierarchy, which is later pruned. Slugger significantly accelerates this process by sampling, approximation, and memoization. Our experiments on 16 real-world graphs show that Slugger is (a) Effective: yielding up to 29.6% more concise summary than state-of-the-art lossless summarization methods, (b) Fast: summarizing a graph with 0.8 billion edges in a few hours, and (c) Scalable: scaling linearly with the number of edges in the input graph.

Post-training compression of Transformer models commonly relies on truncated singular value decomposition (SVD). However, enforcing a single shared subspace can degrade accuracy even at moderate compression. Sparse dictionary learning provides a more flexible union-of-subspaces representation, but existing approaches often suffer from iterative dictionary and coefficient updates. We propose COMPOT (Calibration-Optimized Matrix Procrustes Orthogonalization for Transformers), a training-free compression framework that uses a small calibration dataset to estimate a sparse weight factorization. COMPOT employs orthogonal dictionaries that enable closed-form Procrustes updates for the dictionary and analytical single-step sparse coding for the coefficients, eliminating iterative optimization. To handle heterogeneous layer sensitivity under a global compression budget, COMPOT further introduces a one-shot dynamic allocation strategy that adaptively redistributes layer-wise compression rates. Extensive experiments across diverse architectures and tasks show that COMPOT consistently delivers a superior quality-compression trade-off over strong low-rank and sparse baselines, while remaining fully compatible with post-training quantization for extreme compression. Code is available https://github.com/mts-ai/COMPOT{here}.

Efficient k-nearest neighbor search is a fundamental task, foundational for many problems in NLP. When the similarity is measured by dot-product between dual-encoder vectors or ell_2-distance, there already exist many scalable and efficient search methods. But not so when similarity is measured by more accurate and expensive black-box neural similarity models, such as cross-encoders, which jointly encode the query and candidate neighbor. The cross-encoders' high computational cost typically limits their use to reranking candidates retrieved by a cheaper model, such as dual encoder or TF-IDF. However, the accuracy of such a two-stage approach is upper-bounded by the recall of the initial candidate set, and potentially requires additional training to align the auxiliary retrieval model with the cross-encoder model. In this paper, we present an approach that avoids the use of a dual-encoder for retrieval, relying solely on the cross-encoder. Retrieval is made efficient with CUR decomposition, a matrix decomposition approach that approximates all pairwise cross-encoder distances from a small subset of rows and columns of the distance matrix. Indexing items using our approach is computationally cheaper than training an auxiliary dual-encoder model through distillation. Empirically, for k > 10, our approach provides test-time recall-vs-computational cost trade-offs superior to the current widely-used methods that re-rank items retrieved using a dual-encoder or TF-IDF.

Existing inverse rendering combined with neural rendering methods can only perform editable novel view synthesis on object-specific scenes, while we present intrinsic neural radiance fields, dubbed IntrinsicNeRF, which introduce intrinsic decomposition into the NeRF-based neural rendering method and can extend its application to room-scale scenes. Since intrinsic decomposition is a fundamentally under-constrained inverse problem, we propose a novel distance-aware point sampling and adaptive reflectance iterative clustering optimization method, which enables IntrinsicNeRF with traditional intrinsic decomposition constraints to be trained in an unsupervised manner, resulting in multi-view consistent intrinsic decomposition results. To cope with the problem that different adjacent instances of similar reflectance in a scene are incorrectly clustered together, we further propose a hierarchical clustering method with coarse-to-fine optimization to obtain a fast hierarchical indexing representation. It supports compelling real-time augmented applications such as recoloring and illumination variation. Extensive experiments and editing samples on both object-specific/room-scale scenes and synthetic/real-word data demonstrate that we can obtain consistent intrinsic decomposition results and high-fidelity novel view synthesis even for challenging sequences.

Linear time-invariant state space models (SSM) are a classical model from engineering and statistics, that have recently been shown to be very promising in machine learning through the Structured State Space sequence model (S4). A core component of S4 involves initializing the SSM state matrix to a particular matrix called a HiPPO matrix, which was empirically important for S4's ability to handle long sequences. However, the specific matrix that S4 uses was actually derived in previous work for a particular time-varying dynamical system, and the use of this matrix as a time-invariant SSM had no known mathematical interpretation. Consequently, the theoretical mechanism by which S4 models long-range dependencies actually remains unexplained. We derive a more general and intuitive formulation of the HiPPO framework, which provides a simple mathematical interpretation of S4 as a decomposition onto exponentially-warped Legendre polynomials, explaining its ability to capture long dependencies. Our generalization introduces a theoretically rich class of SSMs that also lets us derive more intuitive S4 variants for other bases such as the Fourier basis, and explains other aspects of training S4, such as how to initialize the important timescale parameter. These insights improve S4's performance to 86% on the Long Range Arena benchmark, with 96% on the most difficult Path-X task.

We study the sparse recovery problem with an underdetermined linear system characterized by a Kronecker-structured dictionary and a Kronecker-supported sparse vector. We cast this problem into the sparse Bayesian learning (SBL) framework and rely on the expectation-maximization method for a solution. To this end, we model the Kronecker-structured support with a hierarchical Gaussian prior distribution parameterized by a Kronecker-structured hyperparameter, leading to a non-convex optimization problem. The optimization problem is solved using the alternating minimization (AM) method and a singular value decomposition (SVD)-based method, resulting in two algorithms. Further, we analytically guarantee that the AM-based method converges to the stationary point of the SBL cost function. The SVD-based method, though it adopts approximations, is empirically shown to be more efficient and accurate. We then apply our algorithm to estimate the uplink wireless channel in an intelligent reflecting surface-aided MIMO system and extend the AM-based algorithm to address block sparsity in the channel. We also study the SBL cost to show that the minima of the cost function are achieved at sparse solutions and that incorporating the Kronecker structure reduces the number of local minima of the SBL cost function. Our numerical results demonstrate the effectiveness of our algorithms compared to the state-of-the-art.

We present a novel global compression framework for deep neural networks that automatically analyzes each layer to identify the optimal per-layer compression ratio, while simultaneously achieving the desired overall compression. Our algorithm hinges on the idea of compressing each convolutional (or fully-connected) layer by slicing its channels into multiple groups and decomposing each group via low-rank decomposition. At the core of our algorithm is the derivation of layer-wise error bounds from the Eckart Young Mirsky theorem. We then leverage these bounds to frame the compression problem as an optimization problem where we wish to minimize the maximum compression error across layers and propose an efficient algorithm towards a solution. Our experiments indicate that our method outperforms existing low-rank compression approaches across a wide range of networks and data sets. We believe that our results open up new avenues for future research into the global performance-size trade-offs of modern neural networks. Our code is available at https://github.com/lucaslie/torchprune.

Second-order optimizers, maintaining a matrix termed a preconditioner, are superior to first-order optimizers in both theory and practice. The states forming the preconditioner and its inverse root restrict the maximum size of models trained by second-order optimizers. To address this, compressing 32-bit optimizer states to lower bitwidths has shown promise in reducing memory usage. However, current approaches only pertain to first-order optimizers. In this paper, we propose the first 4-bit second-order optimizers, exemplified by 4-bit Shampoo, maintaining performance similar to that of 32-bit ones. We show that quantizing the eigenvector matrix of the preconditioner in 4-bit Shampoo is remarkably better than quantizing the preconditioner itself both theoretically and experimentally. By rectifying the orthogonality of the quantized eigenvector matrix, we enhance the approximation of the preconditioner's eigenvector matrix, which also benefits the computation of its inverse 4-th root. Besides, we find that linear square quantization slightly outperforms dynamic tree quantization when quantizing second-order optimizer states. Evaluation on various networks for image classification demonstrates that our 4-bit Shampoo achieves comparable test accuracy to its 32-bit counterpart while being more memory-efficient. The source code will be made available.

Fine-tuning large language models (LMs) for individual tasks yields strong performance but is expensive for deployment and storage. Recent works explore model merging to combine multiple task-specific models into a single multi-task model without additional training. However, existing merging methods often fail for models fine-tuned with low-rank adaptation (LoRA), due to significant performance degradation. In this paper, we show that this issue arises from a previously overlooked interplay between model parameters and data distributions. We propose Orthogonal Subspaces for Robust model Merging (OSRM) to constrain the LoRA subspace *prior* to fine-tuning, ensuring that updates relevant to one task do not adversely shift outputs for others. Our approach can seamlessly integrate with most existing merging algorithms, reducing the unintended interference among tasks. Extensive experiments on eight datasets, tested with three widely used LMs and two large LMs, demonstrate that our method not only boosts merging performance but also preserves single-task accuracy. Furthermore, our approach exhibits greater robustness to the hyperparameters of merging. These results highlight the importance of data-parameter interaction in model merging and offer a plug-and-play solution for merging LoRA models.

We present Midicoth, a lossless compression system that introduces a micro-diffusion denoising layer for improving probability estimates produced by adaptive statistical models. In compressors such as Prediction by Partial Matching (PPM), probability estimates are smoothed by a prior to handle sparse observations. When contexts have been seen only a few times, this prior dominates the prediction and produces distributions that are significantly flatter than the true source distribution, leading to compression inefficiency. Midicoth addresses this limitation by treating prior smoothing as a shrinkage process and applying a reverse denoising step that corrects predicted probabilities using empirical calibration statistics. To make this correction data-efficient, the method decomposes each byte prediction into a hierarchy of binary decisions along a bitwise tree. This converts a single 256-way calibration problem into a sequence of binary calibration tasks, enabling reliable estimation of correction terms from relatively small numbers of observations. The denoising process is applied in multiple successive steps, allowing each stage to refine residual prediction errors left by the previous one. The micro-diffusion layer operates as a lightweight post-blend calibration stage applied after all model predictions have been combined, allowing it to correct systematic biases in the final probability distribution. Midicoth combines five fully online components: an adaptive PPM model, a long-range match model, a trie-based word model, a high-order context model, and the micro-diffusion denoiser applied as the final stage.

Sparse Mixture-of-Experts (SMoE) models represent a significant advancement in large language model (LLM) development through their efficient parameter utilization. These models achieve substantial performance improvements at reduced inference costs. However, the deployment of SMoE models faces constraints from extensive memory requirements of expert components in resource-limited environments. To address these limitations, this paper introduces Hierarchical Clustering for Sparsely activated Mixture of Experts (HC-SMoE), a task-agnostic expert merging framework for parameter reduction without retraining. HC-SMoE introduces a novel hierarchical clustering approach based on expert outputs to ensure merging robustness independent of routing decisions. The proposed output-based clustering method enables effective capture of functional relationships between experts for large-scale architectures. We provide theoretical analysis and comprehensive evaluations across multiple zero-shot language tasks to demonstrate HC-SMoE's effectiveness in state-of-the-art models including Qwen and Mixtral. The experimental results validate HC-SMoE's superior performance and practical applicability for real-world deployments.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

The Mixture-of-Experts (MoE) architecture has become a predominant paradigm for scaling large language models (LLMs). Despite offering strong performance and computational efficiency, large MoE-based LLMs like DeepSeek-V3-0324 and Kimi-K2-Instruct present serious challenges due to substantial memory requirements in deployment. While recent works have explored MoE compression to address this issue, existing methods often suffer from considerable accuracy drops (e.g., 7-14% relatively) even at modest compression rates. This paper introduces a novel Mixture-of-Basis-Experts (MoBE) method that achieves model compression while incurring minimal accuracy drops. Specifically, each up/gate matrix in an expert is decomposed via a rank decomposition as W = AB, where matrix A is unique to each expert. The relatively larger matrix B is further re-parameterized as a linear combination of basis matrices {Bi} shared across all experts within a given MoE layer. The factorization is learned by minimizing the reconstruction error relative to the original weight matrices. Experiments demonstrate that MoBE achieves notably lower accuracy drops compared to prior works. For instance, MoBE can reduce the parameter counts of Qwen3-235B-A22B-2507, DeepSeek-V3-0324 (671B) and Kimi-K2-Instruct (1T) by 24%-30% with only 1%-2% accuracy drop (about 2% drops when measured relatively).

A heuristic extending the Squarified Treemap technique for the representation of hierarchical information as treemaps is presented. The original technique gives high quality treemap views, since items are laid out with rectangles that approximate squares, allowing easy comparison and selection operations. New key steps, with a low computational impact, have been introduced to yield treemaps with even better aspect ratios and higher homogeneity among items.

Time series forecasting is widely used in extensive applications, such as traffic planning and weather forecasting. However, real-world time series usually present intricate temporal variations, making forecasting extremely challenging. Going beyond the mainstream paradigms of plain decomposition and multiperiodicity analysis, we analyze temporal variations in a novel view of multiscale-mixing, which is based on an intuitive but important observation that time series present distinct patterns in different sampling scales. The microscopic and the macroscopic information are reflected in fine and coarse scales respectively, and thereby complex variations can be inherently disentangled. Based on this observation, we propose TimeMixer as a fully MLP-based architecture with Past-Decomposable-Mixing (PDM) and Future-Multipredictor-Mixing (FMM) blocks to take full advantage of disentangled multiscale series in both past extraction and future prediction phases. Concretely, PDM applies the decomposition to multiscale series and further mixes the decomposed seasonal and trend components in fine-to-coarse and coarse-to-fine directions separately, which successively aggregates the microscopic seasonal and macroscopic trend information. FMM further ensembles multiple predictors to utilize complementary forecasting capabilities in multiscale observations. Consequently, TimeMixer is able to achieve consistent state-of-the-art performances in both long-term and short-term forecasting tasks with favorable run-time efficiency.

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.