Daily Papers

NVFP4 is an attractive datatype for large language model (LLM) inference, offering compact storage and native tensor-core acceleration. However, preserving accuracy using NVFP4 requires careful quantization. In this work we develop a novel quantization scheme called Optimized Smoothing and Scaling for NVFP4 (OSFP4). For each linear projection it uses a diagonal smoothing matrix whose entries are optimized to minimize the squared matrix-product quantization error under NVFP4, taking into account the rounding procedure that is used (either round-to-nearest, or GPTQ-style successive interference cancellation). This requires performing joint optimization on the smoothing entries as well as the block scales, which is facilitated by analyzing a multiplicative-dither FP4 quantizer instead of the fixed deterministic one. Experiments show that OSFP4 achieves the highest average accuracy among the evaluated competitors in the corresponding quantization settings, while retaining approximately 94-97\% of vendor NVFP4 prefill throughput on the measured workloads. Our code is available in https://github.com/neriahbd/OSFP4

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices lauritzen2019maximum,slawski2015estimation and even one observation for diagonally dominant M-matrices truell2021maximum. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted ell_1-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

In this paper, we propose new randomized algorithms for estimating the two-to-infinity and one-to-two norms in a matrix-free setting, using only matrix-vector multiplications. Our methods are based on appropriate modifications of Hutchinson's diagonal estimator and its Hutch++ version. We provide oracle complexity bounds for both modifications. We further illustrate the practical utility of our algorithms for Jacobian-based regularization in deep neural network training on image classification tasks. We also demonstrate that our methodology can be applied to mitigate the effect of adversarial attacks in the domain of recommender systems.

Let Ω_n denote the class of n times n doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in Ω_n. The main question is: which A in Ω_n are such that the diagonals in A that avoid the zeros of A all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.

Adaptive rounding methods such as GPTQ, or equivalently Babai's nearest plane algorithm, round a real matrix to integers under a quadratic metric. They process the entries in a fixed order, one at a time, propagating each rounding error to the entries not yet processed through a triangular feedback matrix. We study the two-sided version of this task, in which fixed nonsingular basis matrices act on both the left and the right of the residual; the familiar one-sided case is the special case of an identity right basis. Vectorizing the matrix turns the two-sided objective into a quadratic metric whose Gram matrix is a Kronecker product, so the one-dimensional algorithm applies verbatim, but takes quartic time in the matrix dimension. We present GPTQ-2D, which produces the identical rounded matrix in cubic time. It rounds the entries anti-diagonal by anti-diagonal; entries on the same anti-diagonal are independent and are rounded in parallel.

The probabilistic diffusion model has become highly effective across various domains. Typically, sampling from a diffusion model involves using a denoising distribution characterized by a Gaussian with a learned mean and either fixed or learned covariances. In this paper, we leverage the recently proposed covariance moment matching technique and introduce a novel method for learning the diagonal covariance. Unlike traditional data-driven diagonal covariance approximation approaches, our method involves directly regressing the optimal diagonal analytic covariance using a new, unbiased objective named Optimal Covariance Matching (OCM). This approach can significantly reduce the approximation error in covariance prediction. We demonstrate how our method can substantially enhance the sampling efficiency, recall rate and likelihood of commonly used diffusion models.

In recent years, multiple notions of algorithmic fairness have arisen. One such notion is individual fairness (IF), which requires that individuals who are similar receive similar treatment. In parallel, matrix estimation (ME) has emerged as a natural paradigm for handling noisy data with missing values. In this work, we connect the two concepts. We show that pre-processing data using ME can improve an algorithm's IF without sacrificing performance. Specifically, we show that using a popular ME method known as singular value thresholding (SVT) to pre-process the data provides a strong IF guarantee under appropriate conditions. We then show that, under analogous conditions, SVT pre-processing also yields estimates that are consistent and approximately minimax optimal. As such, the ME pre-processing step does not, under the stated conditions, increase the prediction error of the base algorithm, i.e., does not impose a fairness-performance trade-off. We verify these results on synthetic and real data.

Adaptive moment methods such as Adam use a diagonal, coordinate-wise preconditioner based on exponential moving averages of squared gradients. This diagonal scaling is coordinate-system dependent and can struggle with dense or rotated parameter couplings, including those in matrix factorization, tensor decomposition, and graph neural networks, because it treats each parameter independently. We introduce FlowAdam, a hybrid optimizer that augments Adam with continuous gradient-flow integration via an ordinary differential equation (ODE). When EMA-based statistics detect landscape difficulty, FlowAdam switches to clipped ODE integration. Our central contribution is Soft Momentum Injection, which blends ODE velocity with Adam's momentum during mode transitions. This prevents the training collapse observed with naive hybrid approaches. Across coupled optimization benchmarks, the ODE integration provides implicit regularization, reducing held-out error by 10-22% on low-rank matrix/tensor recovery and 6% on Jester (real-world collaborative filtering), also surpassing tuned Lion and AdaBelief, while matching Adam on well-conditioned workloads (CIFAR-10). MovieLens-100K confirms benefits arise specifically from coupled parameter interactions rather than bias estimation. Ablation studies show that soft injection is essential, as hard replacement reduces accuracy from 100% to 82.5%.

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

Shampoo is an online and stochastic optimization algorithm belonging to the AdaGrad family of methods for training neural networks. It constructs a block-diagonal preconditioner where each block consists of a coarse Kronecker product approximation to full-matrix AdaGrad for each parameter of the neural network. In this work, we provide a complete description of the algorithm as well as the performance optimizations that our implementation leverages to train deep networks at-scale in PyTorch. Our implementation enables fast multi-GPU distributed data-parallel training by distributing the memory and computation associated with blocks of each parameter via PyTorch's DTensor data structure and performing an AllGather primitive on the computed search directions at each iteration. This major performance enhancement enables us to achieve at most a 10% performance reduction in per-step wall-clock time compared against standard diagonal-scaling-based adaptive gradient methods. We validate our implementation by performing an ablation study on training ImageNet ResNet50, demonstrating Shampoo's superiority over standard training recipes with minimal hyperparameter tuning.

We deal with the problem of gradient estimation for stochastic differentiable relaxations of algorithms, operators, simulators, and other non-differentiable functions. Stochastic smoothing conventionally perturbs the input of a non-differentiable function with a differentiable density distribution with full support, smoothing it and enabling gradient estimation. Our theory starts at first principles to derive stochastic smoothing with reduced assumptions, without requiring a differentiable density nor full support, and we present a general framework for relaxation and gradient estimation of non-differentiable black-box functions f:R^ntoR^m. We develop variance reduction for gradient estimation from 3 orthogonal perspectives. Empirically, we benchmark 6 distributions and up to 24 variance reduction strategies for differentiable sorting and ranking, differentiable shortest-paths on graphs, differentiable rendering for pose estimation, as well as differentiable cryo-ET simulations.

In a previous paper it was shown that a machine learning regression problem can be solved within the framework of random function theory, with the optimal kernel analytically derived from symmetry and indifference principles and coinciding with a polyharmonic spline. However, a direct application of that solution is limited by O(N^3) computational cost and by a breakdown of the original theoretical assumptions when the input space has excessive dimensionality. This paper proposes a cascade architecture built from packages of polyharmonic splines that simultaneously addresses scalability and is theoretically justified for problems with unknown intrinsic low dimensionality. Efficient matrix procedures are presented for forward computation and end-to-end differentiation through the cascade.

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.

Linear transformation of the inputs alters the training performance of feed-forward networks that are otherwise equivalent. However, most linear transforms are viewed as a pre-processing operation separate from the actual training. Starting from equivalent networks, it is shown that pre-processing inputs using linear transformation are equivalent to multiplying the negative gradient matrix with an autocorrelation matrix per training iteration. Second order method is proposed to find the autocorrelation matrix that maximizes learning in a given iteration. When the autocorrelation matrix is diagonal, the method optimizes input gains. This optimal input gain (OIG) approach is used to improve two first-order two-stage training algorithms, namely back-propagation (BP) and hidden weight optimization (HWO), which alternately update the input weights and solve linear equations for output weights. Results show that the proposed OIG approach greatly enhances the performance of the first-order algorithms, often allowing them to rival the popular Levenberg-Marquardt approach with far less computation. It is shown that HWO is equivalent to BP with Whitening transformation applied to the inputs. HWO effectively combines Whitening transformation with learning. Thus, OIG improved HWO could be a significant building block to more complex deep learning architectures.

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning.

We propose a novel inference technique based on a pretrained diffusion model for text-conditional video generation. Our approach, called FIFO-Diffusion, is conceptually capable of generating infinitely long videos without training. This is achieved by iteratively performing diagonal denoising, which concurrently processes a series of consecutive frames with increasing noise levels in a queue; our method dequeues a fully denoised frame at the head while enqueuing a new random noise frame at the tail. However, diagonal denoising is a double-edged sword as the frames near the tail can take advantage of cleaner ones by forward reference but such a strategy induces the discrepancy between training and inference. Hence, we introduce latent partitioning to reduce the training-inference gap and lookahead denoising to leverage the benefit of forward referencing. We have demonstrated the promising results and effectiveness of the proposed methods on existing text-to-video generation baselines.

Linear Recurrent Neural Networks (linear RNNs) have emerged as competitive alternatives to Transformers for sequence modeling, offering efficient training and linear-time inference. However, existing architectures face a fundamental trade-off between expressivity and efficiency, dictated by the structure of their state-transition matrices. Diagonal matrices, used in models such as Mamba, GLA, or mLSTM, yield fast runtime but have limited expressivity. To address this, recent architectures such as DeltaNet and RWKV-7 adopted a diagonal plus rank-1 structure, which allows simultaneous token and channel mixing, improving associative recall and, as recently shown, state-tracking when allowing negative eigenvalues in the state-transition matrices. Building on the interpretation of DeltaNet's recurrence as performing one step of online gradient descent per token on an associative recall loss, we introduce DeltaProduct, which instead takes multiple (n_h) steps per token. This naturally leads to diagonal plus rank-n_h state-transition matrices, formed as products of n_h generalized Householder transformations, providing a tunable mechanism to balance expressivity and efficiency. We provide a detailed theoretical characterization of the state-tracking capability of DeltaProduct in finite precision, showing how it improves by increasing n_h. Our extensive experiments demonstrate that DeltaProduct outperforms DeltaNet in both state-tracking and language modeling, while also showing significantly improved length extrapolation capabilities.

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we propose a general method to solve the localization problem by learning a regularization matrix from the localized eigenvectors. Using matrix perturbation analysis, we demonstrate that the learned regularizations suppress down the eigenvalues associated with localized eigenvectors and enable us to recover the informative eigenvectors representing the global structure. We show applications of our method in several inference problems: community detection in networks, clustering from pairwise similarities, rank estimation and matrix completion problems. Using extensive experiments, we illustrate that our method solves the localization problem and works down to the theoretical detectability limits in different kinds of synthetic data. This is in contrast with existing spectral algorithms based on data matrix, non-backtracking matrix, Laplacians and those with rank-one regularizations, which perform poorly in the sparse case with noise.

Randomized Smoothing (RS) has been proven a promising method for endowing an arbitrary image classifier with certified robustness. However, the substantial uncertainty inherent in the high-dimensional isotropic Gaussian noise imposes the curse of dimensionality on RS. Specifically, the upper bound of {ell_2} certified robustness radius provided by RS exhibits a diminishing trend with the expansion of the input dimension d, proportionally decreasing at a rate of 1/d. This paper explores the feasibility of providing {ell_2} certified robustness for high-dimensional input through the utilization of dual smoothing in the lower-dimensional space. The proposed Dual Randomized Smoothing (DRS) down-samples the input image into two sub-images and smooths the two sub-images in lower dimensions. Theoretically, we prove that DRS guarantees a tight {ell_2} certified robustness radius for the original input and reveal that DRS attains a superior upper bound on the {ell_2} robustness radius, which decreases proportionally at a rate of (1/sqrt m + 1/sqrt n ) with m+n=d. Extensive experiments demonstrate the generalizability and effectiveness of DRS, which exhibits a notable capability to integrate with established methodologies, yielding substantial improvements in both accuracy and {ell_2} certified robustness baselines of RS on the CIFAR-10 and ImageNet datasets. Code is available at https://github.com/xiasong0501/DRS.

The rapid growth of model scale has necessitated substantial computational resources for fine-tuning. Existing approach such as Low-Rank Adaptation (LoRA) has sought to address the problem of handling the large updated parameters in full fine-tuning. However, LoRA utilize random initialization and optimization of low-rank matrices to approximate updated weights, which can result in suboptimal convergence and an accuracy gap compared to full fine-tuning. To address these issues, we propose LoLDU, a Parameter-Efficient Fine-Tuning (PEFT) approach that significantly reduces trainable parameters by 2600 times compared to regular PEFT methods while maintaining comparable performance. LoLDU leverages Lower-Diag-Upper Decomposition (LDU) to initialize low-rank matrices for faster convergence and orthogonality. We focus on optimizing the diagonal matrix for scaling transformations. To the best of our knowledge, LoLDU has the fewest parameters among all PEFT approaches. We conducted extensive experiments across 4 instruction-following datasets, 6 natural language understanding (NLU) datasets, 8 image classification datasets, and image generation datasets with multiple model types (LLaMA2, RoBERTa, ViT, and Stable Diffusion), providing a comprehensive and detailed analysis. Our open-source code can be accessed at https://github.com/SKDDJ/LoLDU{https://github.com/SKDDJ/LoLDU}.

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. Y is to be predicted upon observing a (possibly high dimensional) random vector X by means of a predictive function f(X) as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function m(x)=E[Ymid X=x]. Under classic smoothness conditions combined with the assumption that the tails of Y-m(X) are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length o(n^{1/2-ε}) behave like independent exponentials. We determine the limiting empirical distribution of the singular values the the matrix. Finally the mixing time of the associated Markov chains is shown to be exactly 2 with high probability.

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

Diffusion models have achieved remarkable progress in various domains with an intriguing ability to produce new data that do not exist in the training set. In this work, we study the hypothesis that such creativity arises from the neural network backbone learning a smoothed version of the empirical score function, which guides the denoising dynamics to generate data points that interpolate the training data. Focusing mainly on settings where the training set lies uniformly in a one-dimensional subspace, we elucidate the interplay between score smoothing and the denoising dynamics with analytical solutions and numerical experiments, demonstrating how smoothing the score function can cause the denoised data samples to interpolate the training set along the subspace. Moreover, we present theoretical and empirical evidence that learning score functions with neural networks - either with or without explicit regularization - can naturally achieve a similar effect, including when the data belong to simple nonlinear manifolds.

Breakthroughs in machine learning (ML) and advances in quantum computing (QC) drive the interdisciplinary field of quantum machine learning to new levels. However, due to the susceptibility of ML models to adversarial attacks, practical use raises safety-critical concerns. Existing Randomized Smoothing (RS) certification methods for classical machine learning models are computationally intensive. In this paper, we propose the combination of QC and the concept of discrete randomized smoothing to speed up the stochastic certification of ML models for discrete data. We show how to encode all the perturbations of the input binary data in superposition and use Quantum Amplitude Estimation (QAE) to obtain a quadratic reduction in the number of calls to the model that are required compared to traditional randomized smoothing techniques. In addition, we propose a new binary threat model to allow for an extensive evaluation of our approach on images, graphs, and text.

For learning with noisy labels, the transition matrix, which explicitly models the relation between noisy label distribution and clean label distribution, has been utilized to achieve the statistical consistency of either the classifier or the risk. Previous researches have focused more on how to estimate this transition matrix well, rather than how to utilize it. We propose good utilization of the transition matrix is crucial and suggest a new utilization method based on resampling, coined RENT. Specifically, we first demonstrate current utilizations can have potential limitations for implementation. As an extension to Reweighting, we suggest the Dirichlet distribution-based per-sample Weight Sampling (DWS) framework, and compare reweighting and resampling under DWS framework. With the analyses from DWS, we propose RENT, a REsampling method with Noise Transition matrix. Empirically, RENT consistently outperforms existing transition matrix utilization methods, which includes reweighting, on various benchmark datasets. Our code is available at https://github.com/BaeHeeSun/RENT.

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.

We consider the problem of linear estimation, and establish an extension of the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero but bounded with respect to a matrix norm of Schatten type. We derive simple and explicit formulas for the optimal estimator in the cases of Nuclear and Spectral norms (with the Frobenius case recovering ridge regression). Additionally, we analytically derive the generalization error in multiple random matrix ensembles, and compare with Ridge regression. Finally, we conduct an extensive simulation study, in which we show that the cross-validated Nuclear and Spectral regressors can outperform Ridge in several circumstances.

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.

We propose an efficient method for approximating natural gradient descent in neural networks which we call Kronecker-Factored Approximate Curvature (K-FAC). K-FAC is based on an efficiently invertible approximation of a neural network's Fisher information matrix which is neither diagonal nor low-rank, and in some cases is completely non-sparse. It is derived by approximating various large blocks of the Fisher (corresponding to entire layers) as being the Kronecker product of two much smaller matrices. While only several times more expensive to compute than the plain stochastic gradient, the updates produced by K-FAC make much more progress optimizing the objective, which results in an algorithm that can be much faster than stochastic gradient descent with momentum in practice. And unlike some previously proposed approximate natural-gradient/Newton methods which use high-quality non-diagonal curvature matrices (such as Hessian-free optimization), K-FAC works very well in highly stochastic optimization regimes. This is because the cost of storing and inverting K-FAC's approximation to the curvature matrix does not depend on the amount of data used to estimate it, which is a feature typically associated only with diagonal or low-rank approximations to the curvature matrix.

We employ random matrix theory to establish consistency of generalized cross validation (GCV) for estimating prediction risks of sketched ridge regression ensembles, enabling efficient and consistent tuning of regularization and sketching parameters. Our results hold for a broad class of asymptotically free sketches under very mild data assumptions. For squared prediction risk, we provide a decomposition into an unsketched equivalent implicit ridge bias and a sketching-based variance, and prove that the risk can be globally optimized by only tuning sketch size in infinite ensembles. For general subquadratic prediction risk functionals, we extend GCV to construct consistent risk estimators, and thereby obtain distributional convergence of the GCV-corrected predictions in Wasserstein-2 metric. This in particular allows construction of prediction intervals with asymptotically correct coverage conditional on the training data. We also propose an "ensemble trick" whereby the risk for unsketched ridge regression can be efficiently estimated via GCV using small sketched ridge ensembles. We empirically validate our theoretical results using both synthetic and real large-scale datasets with practical sketches including CountSketch and subsampled randomized discrete cosine transforms.

Objective: Electroencephalography signals are recorded as a multidimensional dataset. We propose a new framework based on the augmented covariance extracted from an autoregressive model to improve motor imagery classification. Methods: From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to extend the standard approach using these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. Results: The augmented covariance matrix performed noticeably better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. Conclusion: The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information, incorporating nonlinear components of the signal through an embedding procedure, which allows the leveraging of dynamical systems algorithms. Significance: These results extend the concepts and the results of the Riemannian distance based classification algorithm.

As machine learned models increase in complexity and expressive power, features of simpler models, such as intrinsic coefficient transparency and control over the shape of the modeled function are lost. On the one edge of the spectrum we have simple linear models that possess coefficient transparency, but have a limited expressive power. On the other edge we have neural networks, that have expressive power that improves with scaling, but are mostly opaque. In this work we develop the spectral neuron concept: a scalar model given by f(x)=λ_k (A_0 + A_1 x + ... + A_n x_n), with learned real symmetric matrices A_0, ..., A_n. The input enters the model through an affine matrix function, but the prediction is obtained by reading one of its eigenvalues. Thus, the model is nonlinear, but the source of nonlinearity is still mathematically explicit. This gives us a useful middle ground: the model can become more expressive as the matrix dimension grows, while retaining coefficient transparency through the learned matrices. For example, extremal eigenvalues yield convex or concave functions, semidefinite constraints on the coefficient matrices impose monotonicity, and the associated eigenspaces characterize local feature influence. We study coefficient transparency, feature-influence bounds, and shape-control properties of this model family, and then test whether it can be learned and scaled in practice. We develop a systematic study of this model family, bringing together spectral results from several mathematical literatures to characterize its expressivity, coefficient transparency, feature influence, and shape-control properties. Code available at https://github.com/alexshtf/spectral_neuron_paper.

We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker η that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker η^* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Frechet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.

We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.

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.

Autonomous systems often must predict the motions of nearby agents from partial and noisy data. This paper asks and answers the question: "can we learn, in real-time, a nonlinear predictive model of another agent's motions?" Our online framework denoises and forecasts such dynamics using a modified sliding-window Hankel Dynamic Mode Decomposition (Hankel-DMD). Partial noisy measurements are embedded into a Hankel matrix, while an associated Page matrix enables singular-value hard thresholding (SVHT) to estimate the effective rank. A Cadzow projection enforces structured low-rank consistency, yielding a denoised trajectory and local noise variance estimates. From this representation, a time-varying Hankel-DMD lifted linear predictor is constructed for multi-step forecasts. The residual analysis provides variance-tracking signals that can support downstream estimators and risk-aware planning. We validate the approach in simulation under Gaussian and heavy-tailed noise, and experimentally on a dynamic crane testbed. Results show that the method achieves stable variance-aware denoising and short-horizon prediction suitable for integration into real-time control frameworks.

This article investigates signal estimation in wireless transmission (i.e., receive beamforming) from the perspective of statistical machine learning, where the transmit signals may be from an integrated sensing and communication system; that is, 1) signals may be not only discrete constellation points but also arbitrary complex values; 2) signals may be spatially correlated. Particular attention is paid to handling various uncertainties such as the uncertainty of the transmit signal covariance, the uncertainty of the channel matrix, the uncertainty of the channel noise covariance, the existence of channel impulse noises, and the limited sample size of pilots. To proceed, a distributionally robust machine learning framework that is insensitive to the above uncertainties is proposed, which reveals that channel estimation is not a necessary operation. For optimal linear estimation, the proposed framework includes several existing beamformers as special cases such as diagonal loading and eigenvalue thresholding. For optimal nonlinear estimation, estimators are limited in reproducing kernel Hilbert spaces and neural network function spaces, and corresponding uncertainty-aware solutions (e.g., kernelized diagonal loading) are derived. In addition, we prove that the ridge and kernel ridge regression methods in machine learning are distributionally robust against diagonal perturbation in feature covariance.

Spectral methods are central to high-dimensional statistics and machine learning, underlying procedures for covariance estimation, matrix denoising, representation learning, clustering, and latent variable modeling. In this work, we focus primarily on spectral estimators for high-dimensional factor models, where leading singular vectors of the data matrix are used to estimate latent structure and covariance parameters. In many modern applications, however, data are collected across related but heterogeneous tasks, studies, domains, or populations. Applying such estimators separately to each task can lead to unstable estimates when sample sizes are limited relative to dimension, while completely pooling the data can obscure meaningful task-specific structure. We introduce Hierarchical Spectral Shrinkage (HSS), a scalable empirical-Bayes framework for partially pooling task-specific spectral estimators. The method regularizes the leading spectral directions of each task toward a data-adaptivelylearned common basis, while allowing the amount of shrinkage to vary across tasks and spectral components. The proposed estimator arises as the posterior mean in a surrogate Bayesian regression formulation of the empirical singular vectors. For factor models, HSS yields partially pooled estimators of the group-specific loading spaces and covariance matrices and can be combined with different spectral estimators by replacing their empirical singular vectors with hierarchically regularized counterparts. More broadly, the same construction provides a mechanism for extending spectral estimators to collections of related but heterogeneous datasets. We demonstrate substantial improvements in estimation accuracy and out-of-sample performance over separate, fully pooled, and shared-subspace approaches in synthetic experiments and multi-study gene-expression data.

Optimizers that exploit the matrix structure of gradients are central to modern LLM pre-training, with two distinct frontiers: explicit Kronecker-factored preconditioning -- most recently KL-Shampoo, which estimates the preconditioner via KL divergence minimization -- and orthogonalization of the gradient momentum, exemplified by Muon and analyzed as steepest descent under the spectral norm. The two routes are typically developed in isolation. We make a structural observation about KL-Shampoo's Kronecker preconditioners: their eigenvalue spectra exhibit a spike-and-flat shape -- a few dominant eigenvalues followed by an approximately uniform tail -- across layers and training stages, holding exactly under a rank-ρ signal-plus-noise gradient model. We exploit this structure by restricting one of KL-Shampoo's Kronecker factors to a parametric family aligned with the spike-and-flat shape: full spectral structure on a tracked r-dimensional subspace, single shared eigenvalue across the remaining n-r directions. On these directions, we apply orthogonalization. An identity shows that this orthogonalization recovers the algebraic form of full KL-Shampoo's preconditioner. On four pre-training scales (GPT-2 124M / 350M, LLaMA 134M / 450M), Pro-KLShampoo consistently outperforms KL-Shampoo at every subspace rank we test in validation loss, peak per-GPU memory, and wallclock time to reach each loss level.

We show how to turn any classifier that classifies well under Gaussian noise into a new classifier that is certifiably robust to adversarial perturbations under the ell_2 norm. This "randomized smoothing" technique has been proposed recently in the literature, but existing guarantees are loose. We prove a tight robustness guarantee in ell_2 norm for smoothing with Gaussian noise. We use randomized smoothing to obtain an ImageNet classifier with e.g. a certified top-1 accuracy of 49% under adversarial perturbations with ell_2 norm less than 0.5 (=127/255). No certified defense has been shown feasible on ImageNet except for smoothing. On smaller-scale datasets where competing approaches to certified ell_2 robustness are viable, smoothing delivers higher certified accuracies. Our strong empirical results suggest that randomized smoothing is a promising direction for future research into adversarially robust classification. Code and models are available at http://github.com/locuslab/smoothing.

Denoising diffusion probabilistic models (DDPMs) (Ho et al. 2020) have shown impressive results on image and waveform generation in continuous state spaces. Here, we introduce Discrete Denoising Diffusion Probabilistic Models (D3PMs), diffusion-like generative models for discrete data that generalize the multinomial diffusion model of Hoogeboom et al. 2021, by going beyond corruption processes with uniform transition probabilities. This includes corruption with transition matrices that mimic Gaussian kernels in continuous space, matrices based on nearest neighbors in embedding space, and matrices that introduce absorbing states. The third allows us to draw a connection between diffusion models and autoregressive and mask-based generative models. We show that the choice of transition matrix is an important design decision that leads to improved results in image and text domains. We also introduce a new loss function that combines the variational lower bound with an auxiliary cross entropy loss. For text, this model class achieves strong results on character-level text generation while scaling to large vocabularies on LM1B. On the image dataset CIFAR-10, our models approach the sample quality and exceed the log-likelihood of the continuous-space DDPM model.

Analytic Continual Learning (ACL) offers a computationally efficient alternative to gradient-based approaches. Recent ACL methods are based on Recursive Least Squares (RLS) and have achieved the state-of-the-art results compared to other alternatives. However, they falter significantly in Class-Incremental Learning scenarios characterized by Long-Tailed distributions. While the ill-conditioning of the autocorrelation (Gram) matrix is a known limitation of RLS, we demonstrate that class imbalance exacerbates this issue into a distinct spectral pathology: "tail" classes suffer from severe spectral collapse, rendering their subspaces numerically indistinguishable from noise. Standard Ridge Regression (L_2) fails to address this effectively as it applies isotropic regularization - a uniform penalty that is insufficient to stabilize the tail without over-shrinking the head. To address this, we propose Geometry-Spectral Rectification (GSR), a theoretically grounded framework that treats long-tailed learning as a spectral regularization problem. Unlike standard isotropic regularization (Ridge) which uniformly penalizes all eigenvalues, GSR acts as an anisotropic spectral filter, selectively inflating the collapsed eigenvalues of tail classes. We construct a structured, data-dependent spectral perturbation matrix Δ that selectively inflates collapsed tail eigen-directions of the Gram matrix. Theoretical analysis proves that GSR guarantees an improved stable rank for the Gram matrix, ensuring numerical stability. Extensive experiments show that GSR establishes a new state-of-the-art for analytic CIL, offering a superior trade-off between computational efficiency and robust generalization in long-tailed settings.

When rows of an n times d matrix A are given in a stream, we study algorithms for approximating the top eigenvector of the matrix {A}^TA (equivalently, the top right singular vector of A). We consider worst case inputs A but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter R = σ_1(A)^2/σ_2(A)^2 = Ω(1), then there is a randomized algorithm that uses O(h cdot d cdot polylog(d)) bits of space and outputs a unit vector v that has a correlation 1 - O(1/R) with the top eigenvector v_1. Here h denotes the number of heavy rows in the matrix, defined as the rows with Euclidean norm at least |{A}|_F/d cdot operatorname{polylog(d)}. We also provide a lower bound showing that any algorithm using O(hd/R) bits of space can obtain at most 1 - Ω(1/R^2) correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the R = Ω(log n cdot log d) requirement in a recent work of Price and Xun (FOCS 2024). We note that the algorithm of Price and Xun works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in their analysis can be brought down to R = Ω(log^2 d) for arbitrary order streams and R = Ω(log d) for random order streams. The requirement of R = Ω(log d) for random order streams is nearly tight for their analysis as we obtain a simple instance with R = Ω(log d/loglog d) for which their algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector v_1.

A new class of affine scaling matrices for the interior point Newton-type methods is considered to solve the nonlinear systems with simple bounds. We review the essential properties of a scaling matrix and consider several well-known scaling matrices proposed in the literature. We define a new scaling matrix that is the convex combination of these matrices. The proposed scaling matrix inherits those interesting properties of the individual matrices and satisfies additional desired requirements. The numerical experiments demonstrate the superiority of the new scaling matrix in solving several important test problems.

Projection maintenance is one of the core data structure tasks. Efficient data structures for projection maintenance have led to recent breakthroughs in many convex programming algorithms. In this work, we further extend this framework to the Kronecker product structure. Given a constraint matrix {sf A} and a positive semi-definite matrix Win R^{ntimes n} with a sparse eigenbasis, we consider the task of maintaining the projection in the form of {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}, where {sf B}={sf A}(Wotimes I) or {sf B}={sf A}(W^{1/2}otimes W^{1/2}). At each iteration, the weight matrix W receives a low rank change and we receive a new vector h. The goal is to maintain the projection matrix and answer the query {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}h with good approximation guarantees. We design a fast dynamic data structure for this task and it is robust against an adaptive adversary. Following the beautiful and pioneering work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we use tools from differential privacy to reduce the randomness required by the data structure and further improve the running time.

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent work has shown that these costs can be reduced by line search methods that iteratively adjust the stepsize. We propose an alternative approach to stochastic line search by using a new algorithm based on forward step model building. This model building step incorporates second-order information that allows adjusting not only the stepsize but also the search direction. Noting that deep learning model parameters come in groups (layers of tensors), our method builds its model and calculates a new step for each parameter group. This novel diagonalization approach makes the selected step lengths adaptive. We provide convergence rate analysis, and experimentally show that the proposed algorithm achieves faster convergence and better generalization in well-known test problems. More precisely, SMB requires less tuning, and shows comparable performance to other adaptive methods.

Given a matrix Min R^{mtimes n}, the low rank matrix completion problem asks us to find a rank-k approximation of M as UV^top for Uin R^{mtimes k} and Vin R^{ntimes k} by only observing a few entries specified by a set of entries Omegasubseteq [m]times [n]. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli and Sanghavi~jns13 showed that if M has incoherent rows and columns, then alternating minimization provably recovers the matrix M by observing a nearly linear in n number of entries. While the sample complexity has been subsequently improved~glz17, alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time widetilde O(|Omega| k), which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require widetilde O(|Omega| k^2) time.

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

Randomized smoothing (RS) is one of the prominent techniques to ensure the correctness of machine learning models, where point-wise robustness certificates can be derived analytically. While RS is well understood for classification, its application to generative models is unclear, since their outputs are sequences rather than labels. We resolve this by connecting generative outputs to an oracle classification task and showing that RS can still be enabled: the final response can be classified as a discrete action (e.g., service-robot commands in VLAs), as harmful vs. harmless (content moderation or toxicity detection in VLMs), or even applying oracles to cluster answers into semantically equivalent ones. Provided that the error rate for the oracle classifier comparison is bounded, we develop the theory that associates the number of samples with the corresponding robustness radius. We further derive improved scaling laws analytically relating the certified radius and accuracy to the number of samples, showing that the earlier result of 2 to 3 orders of magnitude fewer samples sufficing with minimal loss remains valid even under weaker assumptions. Together, these advances make robustness certification both well-defined and computationally feasible for state-of-the-art VLMs, as validated against recent jailbreak-style adversarial attacks.

Significant recent work has studied the ability of gradient descent to recover a hidden planted direction θ^star in S^{d-1} in different high-dimensional settings, including tensor PCA and single-index models. The key quantity that governs the ability of gradient descent to traverse these landscapes is the information exponent k^star (Ben Arous et al., (2021)), which corresponds to the order of the saddle at initialization in the population landscape. Ben Arous et al., (2021) showed that n gtrsim d^{max(1, k^star-1)} samples were necessary and sufficient for online SGD to recover θ^star, and Ben Arous et al., (2020) proved a similar lower bound for Langevin dynamics. More recently, Damian et al., (2023) showed it was possible to circumvent these lower bounds by running gradient descent on a smoothed landscape, and that this algorithm succeeds with n gtrsim d^{max(1, k^star/2)} samples, which is optimal in the worst case. This raises the question of whether it is possible to achieve the same rate without explicit smoothing. In this paper, we show that Langevin dynamics can succeed with n gtrsim d^{ k^star/2 } samples if one considers the average iterate, rather than the last iterate. The key idea is that the combination of noise-injection and iterate averaging is able to emulate the effect of landscape smoothing. We apply this result to both the tensor PCA and single-index model settings. Finally, we conjecture that minibatch SGD can also achieve the same rate without adding any additional noise.

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

Matrix functions such as square root, inverse roots, and orthogonalization play a central role in preconditioned gradient methods for neural network training. This has motivated the development of iterative algorithms that avoid explicit eigendecompositions and rely primarily on matrix multiplications, making them well suited for modern GPU accelerators. We present PRISM (Polynomial-fitting and Randomized Iterative Sketching for Matrix functions computation), a general framework for accelerating iterative algorithms for computing matrix functions. PRISM combines adaptive polynomial approximation with randomized sketching: at each iteration, it fits a polynomial surrogate to the current spectrum via a sketched least-squares problem, adapting to the instance at hand with minimal overhead. We apply PRISM to accelerate Newton-Schulz-like iterations for matrix square roots and orthogonalization, which are core primitives in machine learning. Unlike prior methods, PRISM requires no explicit spectral bounds or singular value estimates; and it adapts automatically to the evolving spectrum. Empirically, PRISM accelerates training when integrated into Shampoo and Muon optimizers.

Large pretrained diffusion models have significantly enhanced the quality of generated videos, and yet their use in real-time streaming remains limited. Autoregressive models offer a natural framework for sequential frame synthesis but require heavy computation to achieve high fidelity. Diffusion distillation can compress these models into efficient few-step variants, but existing video distillation approaches largely adapt image-specific methods that neglect temporal dependencies. These techniques often excel in image generation but underperform in video synthesis, exhibiting reduced motion coherence, error accumulation over long sequences, and a latency-quality trade-off. We identify two factors that result in these limitations: insufficient utilization of temporal context during step reduction and implicit prediction of subsequent noise levels in next-chunk prediction (i.e., exposure bias). To address these issues, we propose Diagonal Distillation, which operates orthogonally to existing approaches and better exploits temporal information across both video chunks and denoising steps. Central to our approach is an asymmetric generation strategy: more steps early, fewer steps later. This design allows later chunks to inherit rich appearance information from thoroughly processed early chunks, while using partially denoised chunks as conditional inputs for subsequent synthesis. By aligning the implicit prediction of subsequent noise levels during chunk generation with the actual inference conditions, our approach mitigates error propagation and reduces oversaturation in long-range sequences. We further incorporate implicit optical flow modeling to preserve motion quality under strict step constraints. Our method generates a 5-second video in 2.61 seconds (up to 31 FPS), achieving a 277.3x speedup over the undistilled model.

Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian J_{G} of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner J_{G}^* {F}_t J_{G} induced by any {W}-space preconditioner {F}_t is singular, and consequently the standard chain rule cannot be uniquely inverted to map a preconditioned {W}-space direction back to a factor-space update. We cast existing LoRA optimizers in a unified framework parameterized by two choices: (i) which invertible surrogate for J_{G}^* {F}_t J_{G} to use, and (ii) which {F}_t on {W} to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for J_{G}^* J_{G}, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware {F}_t paired with a closed-form factor-space solve at {O}((m+n)r) memory remains underexplored. We propose AdaPreLoRA, which fills this gap by adopting the Adafactor diagonal Kronecker preconditioner {H}_t on {W} and selecting from the resulting factor-space solution family the element minimizing an {H}_t-weighted imbalance between the two factor contributions; by construction, the resulting factor update is the closest LoRA approximation to the preconditioned {W}-space direction under the {H}_t-weighted norm. Across GPT-2 (E2E), Mistral-7B and Qwen2-7B (GLUE, ARC, GSM8K), and diffusion-model personalization, AdaPreLoRA is competitive with or improves over a representative set of LoRA optimizers while keeping peak GPU memory at the LoRA optimizer level.

In this paper, we introduce a novel theoretical framework for multi-task regression, applying random matrix theory to provide precise performance estimations, under high-dimensional, non-Gaussian data distributions. We formulate a multi-task optimization problem as a regularization technique to enable single-task models to leverage multi-task learning information. We derive a closed-form solution for multi-task optimization in the context of linear models. Our analysis provides valuable insights by linking the multi-task learning performance to various model statistics such as raw data covariances, signal-generating hyperplanes, noise levels, as well as the size and number of datasets. We finally propose a consistent estimation of training and testing errors, thereby offering a robust foundation for hyperparameter optimization in multi-task regression scenarios. Experimental validations on both synthetic and real-world datasets in regression and multivariate time series forecasting demonstrate improvements on univariate models, incorporating our method into the training loss and thus leveraging multivariate information.

With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of least squares (LS) problems min_x|b-Ax|_2, where A in R^{mtimes n}, arise in numerous application areas. Overdetermined standard least squares problems can be solved by using mixed precision within the iterative refinement method of Björck, which transforms the least squares problem into an (m+n)times(m+n) ''augmented'' system. It has recently been shown that mixed precision GMRES-based iterative refinement can also be used, in an approach termed GMRES-LSIR. In practice, we often encounter types of least squares problems beyond standard least squares, including weighted least squares (WLS), min_x|D^{1/2}(b-Ax)|_2, where D^{1/2} is a diagonal matrix of weights. In this paper, we discuss a mixed precision FGMRES-WLSIR algorithm for solving WLS problems using two different preconditioners.

Shampoo, a second-order optimization algorithm which uses a Kronecker product preconditioner, has recently garnered increasing attention from the machine learning community. The preconditioner used by Shampoo can be viewed either as an approximation of the Gauss--Newton component of the Hessian or the covariance matrix of the gradients maintained by Adagrad. We provide an explicit and novel connection between the optimal Kronecker product approximation of these matrices and the approximation made by Shampoo. Our connection highlights a subtle but common misconception about Shampoo's approximation. In particular, the square of the approximation used by the Shampoo optimizer is equivalent to a single step of the power iteration algorithm for computing the aforementioned optimal Kronecker product approximation. Across a variety of datasets and architectures we empirically demonstrate that this is close to the optimal Kronecker product approximation. Additionally, for the Hessian approximation viewpoint, we empirically study the impact of various practical tricks to make Shampoo more computationally efficient (such as using the batch gradient and the empirical Fisher) on the quality of Hessian approximation.

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

Hyperspectral image (HSI) restoration aims at recovering clean images from degraded observations and plays a vital role in downstream tasks. Existing model-based methods have limitations in accurately modeling the complex image characteristics with handcraft priors, and deep learning-based methods suffer from poor generalization ability. To alleviate these issues, this paper proposes an unsupervised HSI restoration framework with pre-trained diffusion model (HIR-Diff), which restores the clean HSIs from the product of two low-rank components, i.e., the reduced image and the coefficient matrix. Specifically, the reduced image, which has a low spectral dimension, lies in the image field and can be inferred from our improved diffusion model where a new guidance function with total variation (TV) prior is designed to ensure that the reduced image can be well sampled. The coefficient matrix can be effectively pre-estimated based on singular value decomposition (SVD) and rank-revealing QR (RRQR) factorization. Furthermore, a novel exponential noise schedule is proposed to accelerate the restoration process (about 5times acceleration for denoising) with little performance decrease. Extensive experimental results validate the superiority of our method in both performance and speed on a variety of HSI restoration tasks, including HSI denoising, noisy HSI super-resolution, and noisy HSI inpainting. The code is available at https://github.com/LiPang/HIRDiff.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Sparse linear regression is one of the most basic questions in machine learning and statistics. Here, we are given as input a design matrix X in R^{N times d} and measurements or labels {y} in R^N where {y} = {X} {w}^* + {xi}, and {xi} is the noise in the measurements. Importantly, we have the additional constraint that the unknown signal vector {w}^* is sparse: it has k non-zero entries where k is much smaller than the ambient dimension. Our goal is to output a prediction vector {w} that has small prediction error: 1{N}cdot |{X} {w}^* - {X} {w}|^2_2. Information-theoretically, we know what is best possible in terms of measurements: under most natural noise distributions, we can get prediction error at most epsilon with roughly N = O(k log d/epsilon) samples. Computationally, this currently needs d^{Omega(k)} run-time. Alternately, with N = O(d), we can get polynomial-time. Thus, there is an exponential gap (in the dependence on d) between the two and we do not know if it is possible to get d^{o(k)} run-time and o(d) samples. We give the first generic positive result for worst-case design matrices {X}: For any {X}, we show that if the support of {w}^* is chosen at random, we can get prediction error epsilon with N = poly(k, log d, 1/epsilon) samples and run-time poly(d,N). This run-time holds for any design matrix {X} with condition number up to 2^{poly(d)}. Previously, such results were known for worst-case {w}^*, but only for random design matrices from well-behaved families, matrices that have a very low condition number (poly(log d); e.g., as studied in compressed sensing), or those with special structural properties.

Real-life applications of deep neural networks are hindered by their unsteady predictions when faced with noisy inputs and adversarial attacks. The certified radius in this context is a crucial indicator of the robustness of models. However how to design an efficient classifier with an associated certified radius? Randomized smoothing provides a promising framework by relying on noise injection into the inputs to obtain a smoothed and robust classifier. In this paper, we first show that the variance introduced by the Monte-Carlo sampling in the randomized smoothing procedure estimate closely interacts with two other important properties of the classifier, i.e. its Lipschitz constant and margin. More precisely, our work emphasizes the dual impact of the Lipschitz constant of the base classifier, on both the smoothed classifier and the empirical variance. To increase the certified robust radius, we introduce a different way to convert logits to probability vectors for the base classifier to leverage the variance-margin trade-off. We leverage the use of Bernstein's concentration inequality along with enhanced Lipschitz bounds for randomized smoothing. Experimental results show a significant improvement in certified accuracy compared to current state-of-the-art methods. Our novel certification procedure allows us to use pre-trained models with randomized smoothing, effectively improving the current certification radius in a zero-shot manner.

The traditional kernel density estimator of an unknown density is by construction completely nonparametric, in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability eta(x)=P(Y=1mid X=x), we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of 2eta(x)-1 and apply the method to real datasets.

Robustness is essential for deep neural networks, especially in security-sensitive applications. To this end, randomized smoothing provides theoretical guarantees for certifying robustness against adversarial perturbations. Recently, diffusion models have been successfully employed for randomized smoothing to purify noise-perturbed samples before making predictions with a standard classifier. While these methods excel at small perturbation radii, they struggle with larger perturbations and incur a significant computational overhead during inference compared to classical methods. To address this, we reformulate the generative modeling task along the diffusion trajectories in pixel space as a discriminative task in the latent space. Specifically, we use instance discrimination to achieve consistent representations along the trajectories by aligning temporally adjacent points. After fine-tuning based on the learned representations, our model enables implicit denoising-then-classification via a single prediction, substantially reducing inference costs. We conduct extensive experiments on various datasets and achieve state-of-the-art performance with minimal computation budget during inference. For example, our method outperforms the certified accuracy of diffusion-based methods on ImageNet across all perturbation radii by 5.3% on average, with up to 11.6% at larger radii, while reducing inference costs by 85times on average. Codes are available at: https://github.com/jiachenlei/rRCM.

Moment-based estimation is a theoretically attractive approach to parametric inference, especially when likelihood-based estimation is unavailable, misspecified, or computationally inconvenient. However, the moment equations involve sample averages, which makes moment-based estimation sensitive to outliers. We propose the SGR-GMM algorithm, a robust generalized method of moments (GMM) procedure that uses a spectral gradient reweighting (SGR) primitive to soft-reweight the per-observation gradients during the moment-matching optimization. Our analysis has three layers. First, for a fixed center, the SGR primitive is formulated as an entropy-regularized spectral game between a sample-weight player and a density-matrix player, which is analyzed using classical multiplicative-weights and matrix-multiplicative-weights regret bounds. Second, we establish explicit convergence radius and finite termination bound for the fixed-center updates in the SGR primitive. Third, we prove a local finite-sample parameter estimation error bound with explicit dependence on the contamination fraction, inlier gradient stability, local GMM identification strength, and optimization accuracy. We further specialize the SGR-GMM algorithm to obtain a robust diagonally-weighted GMM (DGMM) estimator for estimating heteroscedastic low-rank Gaussian mixtures observed under additive Gaussian noise and strong contamination. In the numerical experiments, the SGR primitive produces nearly-oracle gradient estimation and the robust DGMM specialization substantially improves over non-robust moment baselines. The code and data are available at https://github.com/liu-lzhang/sgr-gmm.

We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and Adam. The main disadvantage of traditional second order methods is their heavier per iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in ADAHESSIAN, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that ADAHESSIAN achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of Adam. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that ADAHESSIAN: (i) achieves 1.80%/1.45% higher accuracy on ResNets20/32 on Cifar10, and 5.55% higher accuracy on ImageNet as compared to Adam; (ii) outperforms AdamW for transformers by 0.13/0.33 BLEU score on IWSLT14/WMT14 and 2.7/1.0 PPL on PTB/Wikitext-103; (iii) outperforms AdamW for SqueezeBert by 0.41 points on GLUE; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of ADAHESSIAN is comparable to first order methods, and that it exhibits robustness towards its hyperparameters.

We propose ScaledGD(\lambda), a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, ScaledGD(\lambda) starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, ScaledGD(\lambda) is remarkably robust to ill-conditioning compared to vanilla gradient descent (GD) even with overprameterization. Specifically, we show that, under the Gaussian design, ScaledGD(\lambda) converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla GD which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

Previously, methods for learning marginally stable linear dynamical systems either required the transition matrix to be symmetric or incurred regret bounds that scale polynomially with the system's hidden dimension. In this work, we introduce a novel method that overcomes this trade-off, achieving dimension-free regret despite the presence of asymmetric matrices and marginal stability. Our method combines spectral filtering with linear predictors and employs Chebyshev polynomials in the complex plane to construct a novel spectral filtering basis. This construction guarantees sublinear regret in an online learning framework, without relying on any statistical or generative assumptions. Specifically, we prove that as long as the transition matrix has eigenvalues with complex component bounded by 1/poly log T, then our method achieves regret O(T^{9/10}) when compared to the best linear dynamical predictor in hindsight.

In this article, we introduce a novel normalization technique for neural network weight matrices, which we term weight conditioning. This approach aims to narrow the gap between the smallest and largest singular values of the weight matrices, resulting in better-conditioned matrices. The inspiration for this technique partially derives from numerical linear algebra, where well-conditioned matrices are known to facilitate stronger convergence results for iterative solvers. We provide a theoretical foundation demonstrating that our normalization technique smoothens the loss landscape, thereby enhancing convergence of stochastic gradient descent algorithms. Empirically, we validate our normalization across various neural network architectures, including Convolutional Neural Networks (CNNs), Vision Transformers (ViT), Neural Radiance Fields (NeRF), and 3D shape modeling. Our findings indicate that our normalization method is not only competitive but also outperforms existing weight normalization techniques from the literature.

Shampoo is one of the leading approximate second-order optimizers: a variant of it has won the MLCommons AlgoPerf competition, and it has been shown to produce models with lower activation outliers that are easier to compress. Yet, applying Shampoo currently comes at the cost of significant computational slowdown, due to its expensive internal operations. In this paper, we take a significant step to address this shortcoming by proposing \method (for Distributed Accelerated SHampoo), a faster implementation of Distributed Shampoo based on two main new techniques: First, we show that preconditioner blocks can be stacked into 3D tensors to significantly improve GPU utilization; second, we introduce the Newton-DB iteration and the Chebyshev polynomial approximations as novel and faster approaches for computing the inverse matrix roots required by Shampoo. Along with these algorithmic contributions, we provide a first in-depth analysis of how matrix scaling critically affects Shampoo convergence. On the practical side, our GPU-aware implementation achieves up to 4.83times faster optimizer steps compared to the well-optimized Distributed Shampoo, while Newton-DB attains the lowest validation perplexity per iteration among all tested methods. Our code is available at https://github.com/IST-DASLab/DASH.

While 2D diffusion models generate realistic, high-detail images, 3D shape generation methods like Score Distillation Sampling (SDS) built on these 2D diffusion models produce cartoon-like, over-smoothed shapes. To help explain this discrepancy, we show that the image guidance used in Score Distillation can be understood as the velocity field of a 2D denoising generative process, up to the choice of a noise term. In particular, after a change of variables, SDS resembles a high-variance version of Denoising Diffusion Implicit Models (DDIM) with a differently-sampled noise term: SDS introduces noise i.i.d. randomly at each step, while DDIM infers it from the previous noise predictions. This excessive variance can lead to over-smoothing and unrealistic outputs. We show that a better noise approximation can be recovered by inverting DDIM in each SDS update step. This modification makes SDS's generative process for 2D images almost identical to DDIM. In 3D, it removes over-smoothing, preserves higher-frequency detail, and brings the generation quality closer to that of 2D samplers. Experimentally, our method achieves better or similar 3D generation quality compared to other state-of-the-art Score Distillation methods, all without training additional neural networks or multi-view supervision, and providing useful insights into relationship between 2D and 3D asset generation with diffusion models.

We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. When applied to graph Laplacians, this leads to ultra-sparsifiers that in expectation behave as the nearly-optimal ones given by [Kolla-Makarychev-Saberi-Teng STOC`10]. Combining this with the recursive preconditioning framework by [Spielman-Teng STOC`04] and improved embedding algorithms, this leads to algorithms that solve symmetric diagonally dominant linear systems and electrical flow problems in expected time close to mlog^{1/2}n .

Given only a few observed entries from a low-rank matrix X, matrix completion is the problem of imputing the missing entries, and it formalizes a wide range of real-world settings that involve estimating missing data. However, when there are too few observed entries to complete the matrix, what other aspects of the underlying matrix can be reliably recovered? We study one such problem setting, that of "one-sided" matrix completion, where our goal is to recover the right singular vectors of X, even in the regime where recovering the left singular vectors is impossible, which arises when there are more rows than columns and very few observations. We propose a natural algorithm that involves imputing the missing values of the matrix X^TX and show that even with only two observations per row in X, we can provably recover X^TX as long as we have at least Omega(r^2 d log d) rows, where r is the rank and d is the number of columns. We evaluate our algorithm on one-sided recovery of synthetic data and low-coverage genome sequencing. In these settings, our algorithm substantially outperforms standard matrix completion and a variety of direct factorization methods.

We characterize when a banded totally positive matrix admits a positive bidiagonal factorization. In finite dimension such a factorization always exists. For semi-infinite matrices of arbitrary finite bandwidth, existence is equivalent to convergence of a finite family of positive series determined by the Metelmann factors. Their reciprocals are the limiting values of scalar Jacobi continued fractions. We also prove that every bounded banded totally positive matrix can be approximated in operator norm by matrices admitting positive bidiagonal factorizations, through perturbations confined to a fixed initial block. For banded Toeplitz matrices, the negative roots of the polynomial symbol give a global factorization with bounded factors. Ordered partitions of these roots determine explicit mixed-type multiple orthogonal polynomials. Applying the existing spectral Favard theorem, we compute their matrix of positive measures: its support is one interval, and the measure is purely absolutely continuous with an algebraic density of rank one. These formulas also give transition and first-return laws for the associated killed random walks.

This paper studies a machine learning regression problem as a multivariate approximation problem using the framework of the theory of random functions. An ab initio derivation of a regression method is proposed, starting from postulates of indifference. It is shown that if a probability measure on an infinite-dimensional function space possesses natural symmetries (invariance under translation, rotation, scaling, and Gaussianity), then the entire solution scheme, including the kernel form, the type of regularization, and the noise parameterization, follows analytically from these postulates. The resulting kernel coincides with a generalized polyharmonic spline; however, unlike existing approaches, it is not chosen empirically but arises as a consequence of the indifference principle. This result provides a theoretical foundation for a broad class of smoothing and interpolation methods, demonstrating their optimality in the absence of a priori information.

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}.

In this paper, we propose a speed-up approach for subclass discriminant analysis and formulate a novel efficient multi-view solution to it. The speed-up approach is developed based on graph embedding and spectral regression approaches that involve eigendecomposition of the corresponding Laplacian matrix and regression to its eigenvectors. We show that by exploiting the structure of the between-class Laplacian matrix, the eigendecomposition step can be substituted with a much faster process. Furthermore, we formulate a novel criterion for multi-view subclass discriminant analysis and show that an efficient solution for it can be obtained in a similar to the single-view manner. We evaluate the proposed methods on nine single-view and nine multi-view datasets and compare them with related existing approaches. Experimental results show that the proposed solutions achieve competitive performance, often outperforming the existing methods. At the same time, they significantly decrease the training time.

The singular value decomposition (SVD) is a powerful tool in modern numerical linear algebra, which underpins computational methods such as principal component analysis (PCA), low-rank approximations, and randomized algorithms. Many practical scenarios require solving numerous small SVD problems, a regime generally referred to as "batch SVD". Existing programming models can handle this efficiently on parallel CPU architectures, but high-performance solutions for GPUs remain immature. A GPU-oriented batch SVD solver is introduced. This solver exploits the one-sided Jacobi algorithm to exploit fine-grained parallelism, and a number of algorithmic and design optimizations achieve unmatched performance. Starting from a baseline solver, a sequence of optimizations is applied to obtain incremental performance gains. Numerical experiments show that the new solver is robust across problems with different numerical properties, matrix shapes, and arithmetic precisions. Performance benchmarks on both NVIDIA and AMD systems show significant performance speedups over vendor solutions as well as existing open-source solvers.

Iterative self-training (self-distillation) repeatedly refits a model on pseudo-labels generated by its own predictions. We study this procedure in overparameterized linear regression: an initial estimator is trained on noisy labels, and each subsequent iterate is trained on fresh covariates with noiseless pseudo-labels from the previous model. In the high-dimensional regime, we derive deterministic-equivalent recursions for the prediction risk and effective noise across iterations, and prove that the empirical quantities concentrate sharply around these limits. The recursion separates two competing forces: a systematic component that grows with iteration due to progressive signal forgetting, and a stochastic component that decays due to denoising via repeated data-dependent projections. Their interaction yields a U-shaped test-risk curve and an optimal early-stopping time. In spiked covariance models, iteration further acts as an iteration-dependent spectral filter that preserves strong eigendirections while suppressing weaker ones, inducing an implicit form of soft feature selection distinct from ridge regression. Finally, we propose an iterated generalized cross-validation criterion and prove its uniform consistency for estimating the risk along the self-training trajectory, enabling fully data-driven selection of the stopping time and regularization. Experiments on synthetic covariances validate the theory and illustrate the predicted denoising-forgetting trade-off.

We present a simple but effective technique to smooth out textures while preserving the prominent structures. Our method is built upon a key observation -- the coarsest level in a Gaussian pyramid often naturally eliminates textures and summarizes the main image structures. This inspires our central idea for texture filtering, which is to progressively upsample the very low-resolution coarsest Gaussian pyramid level to a full-resolution texture smoothing result with well-preserved structures, under the guidance of each fine-scale Gaussian pyramid level and its associated Laplacian pyramid level. We show that our approach is effective to separate structure from texture of different scales, local contrasts, and forms, without degrading structures or introducing visual artifacts. We also demonstrate the applicability of our method on various applications including detail enhancement, image abstraction, HDR tone mapping, inverse halftoning, and LDR image enhancement.

The Fisher information is a fundamental concept for characterizing the sensitivity of parameters in neural networks. However, leveraging the full observed Fisher information is too expensive for large models, so most methods rely on simple diagonal approximations. While efficient, this approach ignores parameter correlations, often resulting in reduced performance on downstream tasks. In this work, we mitigate these limitations and propose Generalized Fisher-Weighted SVD (GFWSVD), a post-training LLM compression technique that accounts for both diagonal and off-diagonal elements of the Fisher information matrix, providing a more accurate reflection of parameter importance. To make the method tractable, we introduce a scalable adaptation of the Kronecker-factored approximation algorithm for the observed Fisher information. We demonstrate the effectiveness of our method on LLM compression, showing improvements over existing compression baselines. For example, at a 20 compression rate on the MMLU benchmark, our method outperforms FWSVD, which is based on a diagonal approximation of the Fisher information, by 5 percent, SVD-LLM by 3 percent, and ASVD by 6 percent compression rate.

In this paper, we propose a new class of metric for table structure recognition (TSR) evaluation, called grid table similarity (GriTS). Unlike prior metrics, GriTS evaluates the correctness of a predicted table directly in its natural form as a matrix. To create a similarity measure between matrices, we generalize the two-dimensional largest common substructure (2D-LCS) problem, which is NP-hard, to the 2D most similar substructures (2D-MSS) problem and propose a polynomial-time heuristic for solving it. This algorithm produces both an upper and a lower bound on the true similarity between matrices. We show using evaluation on a large real-world dataset that in practice there is almost no difference between these bounds. We compare GriTS to other metrics and empirically validate that matrix similarity exhibits more desirable behavior than alternatives for TSR performance evaluation. Finally, GriTS unifies all three subtasks of cell topology recognition, cell location recognition, and cell content recognition within the same framework, which simplifies the evaluation and enables more meaningful comparisons across different types of TSR approaches. Code will be released at https://github.com/microsoft/table-transformer.

Riemannian geometry has been applied to Brain Computer Interface (BCI) for brain signals classification yielding promising results. Studying electroencephalographic (EEG) signals from their associated covariance matrices allows a mitigation of common sources of variability (electronic, electrical, biological) by constructing a representation which is invariant to these perturbations. While working in Euclidean space with covariance matrices is known to be error-prone, one might take advantage of algorithmic advances in information geometry and matrix manifold to implement methods for Symmetric Positive-Definite (SPD) matrices. This paper proposes a comprehensive review of the actual tools of information geometry and how they could be applied on covariance matrices of EEG. In practice, covariance matrices should be estimated, thus a thorough study of all estimators is conducted on real EEG dataset. As a main contribution, this paper proposes an online implementation of a classifier in the Riemannian space and its subsequent assessment in Steady-State Visually Evoked Potential (SSVEP) experimentations.

Online learning is an attractive framework in many domains because it permits well-defined learning even when data are dependent or chosen adversarially. This generality, however, comes at a steep price, introducing significant statistical and computational barriers. Recently, smoothed online learning has emerged as a promising framework that interpolates between the fully adversarial and fully stochastic settings by assuming that the conditional law of each covariate has density at most 1/σ with respect to some fixed base measure μ, and it is known to match the statistical and computational guarantees of classical learning while still allowing for much of the flexibility of online learning. However, existing oracle-efficient algorithms require either (i) sampling access to the base measure μ or (ii) labels that are perfectly predicted by a fixed hypothesis. Both assumptions limit the applicability of these algorithms, in contrast to statistical learning, where empirical risk minimization (ERM) learns efficiently in the agnostic setting without any knowledge of the data distribution. We show that neither assumption is necessary, giving the first oracle-efficient algorithm that achieves sublinear regret in the agnostic setting without knowledge of μ. Our algorithm, based on Gaussian Follow-The-Perturbed-Leader, is parameter-free: it requires no knowledge of μ, the smoothing parameter σ, or the horizon T, and it achieves regret widetilde O(dT/σ) for binary classes of VC dimension d with a single call to an ERM oracle per round, which is optimal up to a d factor. En route to establishing the regret bound, we introduce several new techniques that may be of independent interest.

Randomized smoothing is the state-of-the-art approach to construct image classifiers that are provably robust against additive adversarial perturbations of bounded magnitude. However, it is more complicated to construct reasonable certificates against semantic transformation (e.g., image blurring, translation, gamma correction) and their compositions. In this work, we propose General Lipschitz (GL), a new framework to certify neural networks against composable resolvable semantic perturbations. Within the framework, we analyze transformation-dependent Lipschitz-continuity of smoothed classifiers w.r.t. transformation parameters and derive corresponding robustness certificates. Our method performs comparably to state-of-the-art approaches on the ImageNet dataset.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

Providing a model that achieves a strong predictive performance and is simultaneously interpretable by humans is one of the most difficult challenges in machine learning research due to the conflicting nature of these two objectives. To address this challenge, we propose a modification of the radial basis function neural network model by equipping its Gaussian kernel with a learnable precision matrix. We show that precious information is contained in the spectrum of the precision matrix that can be extracted once the training of the model is completed. In particular, the eigenvectors explain the directions of maximum sensitivity of the model revealing the active subspace and suggesting potential applications for supervised dimensionality reduction. At the same time, the eigenvectors highlight the relationship in terms of absolute variation between the input and the latent variables, thereby allowing us to extract a ranking of the input variables based on their importance to the prediction task enhancing the model interpretability. We conducted numerical experiments for regression, classification, and feature selection tasks, comparing our model against popular machine learning models, the state-of-the-art deep learning-based embedding feature selection techniques, and a transformer model for tabular data. Our results demonstrate that the proposed model does not only yield an attractive prediction performance compared to the competitors but also provides meaningful and interpretable results that potentially could assist the decision-making process in real-world applications. A PyTorch implementation of the model is available on GitHub at the following link. https://github.com/dannyzx/Gaussian-RBFNN

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

The gradual nature of a diffusion process that synthesizes samples in small increments constitutes a key ingredient of Denoising Diffusion Probabilistic Models (DDPM), which have presented unprecedented quality in image synthesis and been recently explored in the motion domain. In this work, we propose to adapt the gradual diffusion concept (operating along a diffusion time-axis) into the temporal-axis of the motion sequence. Our key idea is to extend the DDPM framework to support temporally varying denoising, thereby entangling the two axes. Using our special formulation, we iteratively denoise a motion buffer that contains a set of increasingly-noised poses, which auto-regressively produces an arbitrarily long stream of frames. With a stationary diffusion time-axis, in each diffusion step we increment only the temporal-axis of the motion such that the framework produces a new, clean frame which is removed from the beginning of the buffer, followed by a newly drawn noise vector that is appended to it. This new mechanism paves the way towards a new framework for long-term motion synthesis with applications to character animation and other domains.

Every Cartesian three-dimensional PDE solver hides a structural secret that production CFD codes have used for half a century and that graduate-level textbooks rarely state plainly. The derivative matrices, the compact Padé line solves, the Galerkin mass inversions, the alternating-direction-implicit substeps, and even the fast Poisson and Helmholtz diagonalization transforms all factor along the coordinate axes and collapse into repeated one-dimensional banded kernels executed along the grid lines. The three-dimensional operator exists only on paper; it is never assembled, factored, or stored. This paper is the manual for that collapse. We derive the Kronecker-product algebra that makes it exact, carry it cleanly through central differences, compact schemes, tensor-product Galerkin, B-spline and isogeometric methods, collocation, ADI time stepping, and direct Poisson and Helmholtz solves, and bring into the open the three production tricks that turn the reduction into hardware-conscious floating-point throughput on real machines: the multi-right-hand-side reshape that exposes a sweep as one batched line kernel (a dense BLAS-3 GEMM when the line factor is dense or element-local, a banded or stencil kernel when it is not), the sum factorization that rescues high-order Galerkin from the O(p^{2d}) quadrature trap, and the pencil decomposition that keeps every direction contiguous across an MPI cluster. For fixed stencil width or fixed polynomial degree, the compute cost stays O(N) in the total number of unknowns N = N_x N_y N_z; the operator storage drops to O(N_x + N_y + N_z) up to bandwidth constants; direct separable Poisson and Helmholtz solvers add the expected transform cost; the line kernels are embarrassingly parallel. These facts are familiar to practitioners but rarely assembled in one place; this paper collects them and shows how to use them.

Despite their remarkable effectiveness and broad application, the drivers of success underlying ensembles of trees are still not fully understood. In this paper, we highlight how interpreting tree ensembles as adaptive and self-regularizing smoothers can provide new intuition and deeper insight to this topic. We use this perspective to show that, when studied as smoothers, randomized tree ensembles not only make predictions that are quantifiably more smooth than the predictions of the individual trees they consist of, but also further regulate their smoothness at test-time based on the dissimilarity between testing and training inputs. First, we use this insight to revisit, refine and reconcile two recent explanations of forest success by providing a new way of quantifying the conjectured behaviors of tree ensembles objectively by measuring the effective degree of smoothing they imply. Then, we move beyond existing explanations for the mechanisms by which tree ensembles improve upon individual trees and challenge the popular wisdom that the superior performance of forests should be understood as a consequence of variance reduction alone. We argue that the current high-level dichotomy into bias- and variance-reduction prevalent in statistics is insufficient to understand tree ensembles -- because the prevailing definition of bias does not capture differences in the expressivity of the hypothesis classes formed by trees and forests. Instead, we show that forests can improve upon trees by three distinct mechanisms that are usually implicitly entangled. In particular, we demonstrate that the smoothing effect of ensembling can reduce variance in predictions due to noise in outcome generation, reduce variability in the quality of the learned function given fixed input data and reduce potential bias in learnable functions by enriching the available hypothesis space.

Sparse matrix factorization is the problem of approximating a matrix Z by a product of J sparse factors X^{(J)} X^{(J-1)} ldots X^{(1)}. This paper focuses on identifiability issues that appear in this problem, in view of better understanding under which sparsity constraints the problem is well-posed. We give conditions under which the problem of factorizing a matrix into two sparse factors admits a unique solution, up to unavoidable permutation and scaling equivalences. Our general framework considers an arbitrary family of prescribed sparsity patterns, allowing us to capture more structured notions of sparsity than simply the count of nonzero entries. These conditions are shown to be related to essential uniqueness of exact matrix decomposition into a sum of rank-one matrices, with structured sparsity constraints. In particular, in the case of fixed-support sparse matrix factorization, we give a general sufficient condition for identifiability based on rank-one matrix completability, and we derive from it a completion algorithm that can verify if this sufficient condition is satisfied, and recover the entries in the two sparse factors if this is the case. A companion paper further exploits these conditions to derive identifiability properties and theoretically sound factorization methods for multi-layer sparse matrix factorization with support constraints associated to some well-known fast transforms such as the Hadamard or the Discrete Fourier Transforms.

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.

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.

Training deep neural networks is a structured optimization problem, because the parameters are naturally represented by matrices and tensors rather than by vectors. Under this structural representation, it has been widely observed that gradients are low-rank and Hessians are approximately block diagonal. These structured properties are crucial for designing efficient optimization algorithms, but are not utilized by many current popular optimizers like Adam. In this paper, we present a novel optimization algorithm ASGO that capitalizes on these properties by employing a preconditioner that is adaptively updated using structured gradients. By a fine-grained theoretical analysis, ASGO is proven to achieve superior convergence rates compared to existing structured gradient methods. Based on this convergence theory, we further demonstrate that ASGO can benefit from low-rank gradients and block diagonal Hessians. We also discuss practical modifications of ASGO and empirically verify ASGO's effectiveness on language model tasks. Code is available at https://github.com/infinity-stars/ASGO.

Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.

Scientific observations are frequently distributed across locations, time periods, and institutions. Combining such observations into a continuous, differentiable field enables recovering governing physical parameters from its derivatives. This paper makes two contributions in this setting. First, the established additive structure of fixed-basis ridge-regression statistics is applied to tensor-product spline fields: each data holder computes a local Gram matrix and moment vector, and the merged solution is mathematically identical to centralized fitting, with no raw data shared and no iterative synchronization. This property is specific to the fixed-feature squared-error setting; the present derivation does not establish an analogous guarantee for general jointly trained multilayer networks. Second, a complete pipeline connects distributed observations to physical parameter inference through field reconstruction, derivative extraction, and linear regression. The pipeline is validated on four PDEs: diffusion, wave, heat-with-source, and the nonlinear viscous Burgers equation, recovering governing parameters to sub-percent accuracy in the linear cases and 5\% for Burgers. In all cases, distributed merging introduces zero degradation relative to centralized fitting. Synthetic experiments validate parameter recovery; application to 41 years of NOAA sea-surface temperature data validates field reconstruction and aggregation equivalence on real spatiotemporal observations. Source code to reproduce all experiments is available at https://github.com/NAVEENMN/splinemerge.

Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs.

Proximal causal learning is a promising framework for identifying the causal effect under the existence of unmeasured confounders. Within this framework, the doubly robust (DR) estimator was derived and has shown its effectiveness in estimation, especially when the model assumption is violated. However, the current form of the DR estimator is restricted to binary treatments, while the treatment can be continuous in many real-world applications. The primary obstacle to continuous treatments resides in the delta function present in the original DR estimator, making it infeasible in causal effect estimation and introducing a heavy computational burden in nuisance function estimation. To address these challenges, we propose a kernel-based DR estimator that can well handle continuous treatments. Equipped with its smoothness, we show that its oracle form is a consistent approximation of the influence function. Further, we propose a new approach to efficiently solve the nuisance functions. We then provide a comprehensive convergence analysis in terms of the mean square error. We demonstrate the utility of our estimator on synthetic datasets and real-world applications.

Large neural networks excel in many domains, but they are expensive to train and fine-tune. A popular approach to reduce their compute or memory requirements is to replace dense weight matrices with structured ones (e.g., sparse, low-rank, Fourier transform). These methods have not seen widespread adoption (1) in end-to-end training due to unfavorable efficiency--quality tradeoffs, and (2) in dense-to-sparse fine-tuning due to lack of tractable algorithms to approximate a given dense weight matrix. To address these issues, we propose a class of matrices (Monarch) that is hardware-efficient (they are parameterized as products of two block-diagonal matrices for better hardware utilization) and expressive (they can represent many commonly used transforms). Surprisingly, the problem of approximating a dense weight matrix with a Monarch matrix, though nonconvex, has an analytical optimal solution. These properties of Monarch matrices unlock new ways to train and fine-tune sparse and dense models. We empirically validate that Monarch can achieve favorable accuracy-efficiency tradeoffs in several end-to-end sparse training applications: speeding up ViT and GPT-2 training on ImageNet classification and Wikitext-103 language modeling by 2x with comparable model quality, and reducing the error on PDE solving and MRI reconstruction tasks by 40%. In sparse-to-dense training, with a simple technique called "reverse sparsification," Monarch matrices serve as a useful intermediate representation to speed up GPT-2 pretraining on OpenWebText by 2x without quality drop. The same technique brings 23% faster BERT pretraining than even the very optimized implementation from Nvidia that set the MLPerf 1.1 record. In dense-to-sparse fine-tuning, as a proof-of-concept, our Monarch approximation algorithm speeds up BERT fine-tuning on GLUE by 1.7x with comparable accuracy.

Diffusion models have achieved remarkable success in text-to-image synthesis, largely attributed to the use of classifier-free guidance (CFG), which enables high-quality, condition-aligned image generation. CFG combines the conditional score (e.g., text-conditioned) with the unconditional score to control the output. However, the unconditional score is in charge of estimating the transition between manifolds of adjacent timesteps from x_t to x_{t-1}, which may inadvertently interfere with the trajectory toward the specific condition. In this work, we introduce a novel approach that leverages a geometric perspective on the unconditional score to enhance CFG performance when conditional scores are available. Specifically, we propose a method that filters the singular vectors of both conditional and unconditional scores using singular value decomposition. This filtering process aligns the unconditional score with the conditional score, thereby refining the sampling trajectory to stay closer to the manifold. Our approach improves image quality with negligible additional computation. We provide deeper insights into the score function behavior in diffusion models and present a practical technique for achieving more accurate and contextually coherent image synthesis.

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of spectral algorithms specified by profile h(lambda), and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile h(lambda) for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

We study the complexity of sampling from a distribution over all index subsets of the set {1,...,n} with the probability of a subset S proportional to the determinant of the submatrix L_S of some ntimes n p.s.d. matrix L, where L_S corresponds to the entries of L indexed by S. Known as a determinantal point process, this distribution is used in machine learning to induce diversity in subset selection. In practice, we often wish to sample multiple subsets S with small expected size k = E[|S|] ll n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose, we propose a new algorithm which, given access to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is n cdot poly(k), i.e., sublinear in the size of L, and (2) its sampling cost is poly(k), i.e., independent of the size of L. Prior to our results, state-of-the-art exact samplers required O(n^3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We also give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with ncdotpoly(k) time preprocessing.

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order frac{klog L{m}}, where m is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

When you merge two fine-tuned models from the same base checkpoint by simply averaging their weights, you implicitly assume their attention subspaces are still aligned. Recent work attempts to fix misalignments by learning an invertible correction matrix, M, for each model's query and key projections. This correction cancels out---using M on the query side and M^{-T} on the key side---right before the dot product. However, this cancellation is only exact if nothing sits between the projection and the dot product. In reality, almost all modern open-weight language models put a rotary position embedding (RoPE) exactly there. In this paper, we show that this cancellation is exact under RoPE if and only if M commutes with RoPE's per-position rotation. We derive the specific class of matrices where this holds: a scaled rotation acting independently within each RoPE frequency pair. This forms a strict, low-dimensional subset of the unconstrained matrices that current methods normally train. Building on this, we turn this constrained matrix class into a new merging method. While keeping the base weights entirely frozen, two fine-tunes each learn their own RoPE-compliant correction matrices. We optimize these corrections against a chosen blend ratio so the final result can be adjusted post-hoc like a dial, rather than locked into a single fixed merge. Our default approach trains at one fixed blend ratio, similar to how LoRA sets its scaling hyperparameter in advance. We also experiment with resampling the blend ratio randomly at every training step, and we report the results of both approaches.

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix K_t. To avoid storing the entire matrix K_t, Nystrom methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed matrix. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, recent works show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for the approximation. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INK-ESTIMATE maintains a small sketch of K_t, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance between the true kernel matrix and its approximation, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.

Predicting the dependencies between observations from multiple time series is critical for applications such as anomaly detection, financial risk management, causal analysis, or demand forecasting. However, the computational and numerical difficulties of estimating time-varying and high-dimensional covariance matrices often limits existing methods to handling at most a few hundred dimensions or requires making strong assumptions on the dependence between series. We propose to combine an RNN-based time series model with a Gaussian copula process output model with a low-rank covariance structure to reduce the computational complexity and handle non-Gaussian marginal distributions. This permits to drastically reduce the number of parameters and consequently allows the modeling of time-varying correlations of thousands of time series. We show on several real-world datasets that our method provides significant accuracy improvements over state-of-the-art baselines and perform an ablation study analyzing the contributions of the different components of our model.

Popular parameter-efficient fine-tuning (PEFT) methods, such as LoRA and its variants, freeze pre-trained model weights \(W\) and inject learnable matrices \(\Delta W\). These \(\Delta W\) matrices are structured for efficient parameterization, often using techniques like low-rank approximations or scaling vectors. However, these methods typically show a performance gap compared to full fine-tuning. Although recent PEFT methods have narrowed this gap, they do so at the cost of additional learnable parameters. We propose SVFT, a simple approach that fundamentally differs from existing methods: the structure imposed on \(\Delta W\) depends on the specific weight matrix \(W\). Specifically, SVFT updates \(W\) as a sparse combination of outer products of its singular vectors, training only the coefficients (scales) of these sparse combinations. This approach allows fine-grained control over expressivity through the number of coefficients. Extensive experiments on language and vision benchmarks show that SVFT recovers up to 96% of full fine-tuning performance while training only 0.006 to 0.25% of parameters, outperforming existing methods that only recover up to 85% performance using 0.03 to 0.8% of the trainable parameter budget.

We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov--Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets A,BsubsetR^d of positive measure with finite upper codimension-one Minkowski boundary content, the number Λ_varepsilon(cA,B) of eigenvalues in (varepsilon,1-varepsilon) satisfies Λ_varepsilon(cA,B)le C(d,A,B,β)c^{d-1}D(1+log_+(c/D)), where D=log(1/(varepsilon(1-varepsilon))), for every fixed β>0, cge2, 0<varepsilon<1/2, and Dleβc, with log_+t=max(0,log t). The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant d-1 for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For C^1 convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is 2. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where M^* in R^{n times n} is a positive semi-definite unknown matrix of rank r ll n, and one uses a symmetric parameterization XX^top to learn M^*. Here X in R^{n times k} with k > r is the factor matrix. We give a novel Omega (1/T^2) lower bound of randomly initialized GD for the over-parameterized case (k >r) where T is the number of iterations. This is in stark contrast to the exact-parameterization scenario (k=r) where the convergence rate is exp (-Omega (T)). Next, we study asymmetric setting where M^* in R^{n_1 times n_2} is the unknown matrix of rank r ll min{n_1,n_2}, and one uses an asymmetric parameterization FG^top to learn M^* where F in R^{n_1 times k} and G in R^{n_2 times k}. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case (k=r) with an exp (-Omega(T)) rate. Furthermore, we give the first global exact convergence result for the over-parameterization case (k>r) with an exp(-Omega(alpha^2 T)) rate where alpha is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from Omega (1/T^2) to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of alpha, recovering the rate in the exact-parameterization case.

The recent advancements in text-to-3D generation mark a significant milestone in generative models, unlocking new possibilities for creating imaginative 3D assets across various real-world scenarios. While recent advancements in text-to-3D generation have shown promise, they often fall short in rendering detailed and high-quality 3D models. This problem is especially prevalent as many methods base themselves on Score Distillation Sampling (SDS). This paper identifies a notable deficiency in SDS, that it brings inconsistent and low-quality updating direction for the 3D model, causing the over-smoothing effect. To address this, we propose a novel approach called Interval Score Matching (ISM). ISM employs deterministic diffusing trajectories and utilizes interval-based score matching to counteract over-smoothing. Furthermore, we incorporate 3D Gaussian Splatting into our text-to-3D generation pipeline. Extensive experiments show that our model largely outperforms the state-of-the-art in quality and training efficiency.

As the use of artificial intelligence rapidly increases, the development of trustworthy artificial intelligence has become important. However, recent studies have shown that deep neural networks are susceptible to learn spurious correlations present in datasets. To improve the reliability, we propose a simple yet effective framework called controllable feature whitening. We quantify the linear correlation between the target and bias features by the covariance matrix, and eliminate it through the whitening module. Our results systemically demonstrate that removing the linear correlations between features fed into the last linear classifier significantly mitigates the bias, while avoiding the need to model intractable higher-order dependencies. A particular advantage of the proposed method is that it does not require regularization terms or adversarial learning, which often leads to unstable optimization in practice. Furthermore, we show that two fairness criteria, demographic parity and equalized odds, can be effectively handled by whitening with the re-weighted covariance matrix. Consequently, our method controls the trade-off between the utility and fairness of algorithms by adjusting the weighting coefficient. Finally, we validate that our method outperforms existing approaches on four benchmark datasets: Corrupted CIFAR-10, Biased FFHQ, WaterBirds, and Celeb-A.

This paper investigates the distinctions between gradient methods applied to non-differentiable functions (NGDMs) and classical gradient descents (GDs) designed for differentiable functions. First, we demonstrate significant differences in the convergence properties of NGDMs compared to GDs, challenging the applicability of the extensive neural network convergence literature based on L-smoothness to non-smooth neural networks. Next, we demonstrate the paradoxical nature of NGDM solutions for L_{1}-regularized problems, showing that increasing the regularization penalty leads to an increase in the L_{1} norm of optimal solutions in NGDMs. Consequently, we show that widely adopted L_{1} penalization-based techniques for network pruning do not yield expected results. Finally, we explore the Edge of Stability phenomenon, indicating its inapplicability even to Lipschitz continuous convex differentiable functions, leaving its relevance to non-convex non-differentiable neural networks inconclusive. Our analysis exposes misguided interpretations of NGDMs in widely referenced papers and texts due to an overreliance on strong smoothness assumptions, emphasizing the necessity for a nuanced understanding of foundational assumptions in the analysis of these systems.

Flow- and diffusion-based generative models have recently emerged as flexible and highly efficient forecasting models for dynamical systems. When combined with inference-time guidance, they offer a promising route to high-dimensional non-Gaussian data assimilation (DA), the problem of combining forecasts with observations to estimate latent system states. Existing filters, however, condition on a fixed history and assimilate only the most recent observation, leaving them unable to revise past states when new observations arrive. Estimates then stay tethered to a history that later observations may contradict, and errors accumulate over the assimilation run. To this end, we introduce **DAWIS**, a unified DA method covering filtering, fixed-lag smoothing, and block smoothing within a single framework. DAWIS replaces the single flow time of a state-level prior with a multitask stochastic interpolant over a window of consecutive states, assigning a separate flow time to each. An assimilation cycle inverts the window to a vector of per-state turning points and regenerates it under observation guidance, with the turning points controlling how strongly each state is held fixed, revised, or generated from scratch. The same construction can also absorb the forecast into the assimilation cycle, removing the need for a separate forecasting model. Experiments on challenging nonlinear systems show that DAWIS improves on both filtering and smoothing baselines under sparse, noisy, and nonlinear observations. The code for DAWIS is available at https://github.com/Erik-Wikingsson/DAWIS

In this paper, we derive a novel bound on the generalization error of Magnitude-Based pruning of overparameterized neural networks. Our work builds on the bounds in Arora et al. [2018] where the error depends on one, the approximation induced by pruning, and two, the number of parameters in the pruned model, and improves upon standard norm-based generalization bounds. The pruned estimates obtained using our new Magnitude-Based compression algorithm are close to the unpruned functions with high probability, which improves the first criteria. Using Sparse Matrix Sketching, the space of the pruned matrices can be efficiently represented in the space of dense matrices of much smaller dimensions, thereby lowering the second criterion. This leads to stronger generalization bound than many state-of-the-art methods, thereby breaking new ground in the algorithm development for pruning and bounding generalization error of overparameterized models. Beyond this, we extend our results to obtain generalization bound for Iterative Pruning [Frankle and Carbin, 2018]. We empirically verify the success of this new method on ReLU-activated Feed Forward Networks on the MNIST and CIFAR10 datasets.