Fundamental Limits of Matrix Sensing: Exact Asymptotics, Universality, and Applications

• URL: http://arxiv.org/abs/2503.14121v1
• Date: Tue, 18 Mar 2025 10:36:30 GMT
• Title: Fundamental Limits of Matrix Sensing: Exact Asymptotics, Universality, and Applications
• Authors: Yizhou Xu, Antoine Maillard, Lenka Zdeborová, Florent Krzakala,
• Abstract summary: We provide rigorous equations characterizing the Bayes-optimal learning performance from a number of samples.<n>We mathematically establish predictions obtained via non-rigorous methods from statistical physics.
• Score: 30.659400341011004
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the matrix sensing problem, one wishes to reconstruct a matrix from (possibly noisy) observations of its linear projections along given directions. We consider this model in the high-dimensional limit: while previous works on this model primarily focused on the recovery of low-rank matrices, we consider in this work more general classes of structured signal matrices with potentially large rank, e.g. a product of two matrices of sizes proportional to the dimension. We provide rigorous asymptotic equations characterizing the Bayes-optimal learning performance from a number of samples which is proportional to the number of entries in the matrix. Our proof is composed of three key ingredients: $(i)$ we prove universality properties to handle structured sensing matrices, related to the ''Gaussian equivalence'' phenomenon in statistical learning, $(ii)$ we provide a sharp characterization of Bayes-optimal learning in generalized linear models with Gaussian data and structured matrix priors, generalizing previously studied settings, and $(iii)$ we leverage previous works on the problem of matrix denoising. The generality of our results allow for a variety of applications: notably, we mathematically establish predictions obtained via non-rigorous methods from statistical physics in [ETB+24] regarding Bilinear Sequence Regression, a benchmark model for learning from sequences of tokens, and in [MTM+24] on Bayes-optimal learning in neural networks with quadratic activation function, and width proportional to the dimension.

Related papers

• Two-Point Deterministic Equivalence for Stochastic Gradient Dynamics in Linear Models [76.52307406752556]
  We derive a novel deterministic equivalence for the two-point function of a random resolvent.<n>We give a unified derivation of the performance of a wide variety of high-dimensional trained linear models with gradient descent.
  arXiv  Detail & Related papers  (2025-02-07T16:45:40Z)
• The Decimation Scheme for Symmetric Matrix Factorization [0.0]
  Matrix factorization is an inference problem that has acquired importance due to its vast range of applications. We study this extensive rank problem, extending the alternative 'decimation' procedure that we recently introduced. We introduce a simple algorithm based on a ground state search that implements decimation and performs matrix factorization.
  arXiv  Detail & Related papers  (2023-07-31T10:53:45Z)
• Learning Graphical Factor Models with Riemannian Optimization [70.13748170371889]
  This paper proposes a flexible algorithmic framework for graph learning under low-rank structural constraints. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution. We leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models.
  arXiv  Detail & Related papers  (2022-10-21T13:19:45Z)
• Graph Polynomial Convolution Models for Node Classification of Non-Homophilous Graphs [52.52570805621925]
  We investigate efficient learning from higher-order graph convolution and learning directly from adjacency matrix for node classification. We show that the resulting model lead to new graphs and residual scaling parameter. We demonstrate that the proposed methods obtain improved accuracy for node-classification of non-homophilous parameters.
  arXiv  Detail & Related papers  (2022-09-12T04:46:55Z)
• Semi-Supervised Subspace Clustering via Tensor Low-Rank Representation [64.49871502193477]
  We propose a novel semi-supervised subspace clustering method, which is able to simultaneously augment the initial supervisory information and construct a discriminative affinity matrix. Comprehensive experimental results on six commonly-used benchmark datasets demonstrate the superiority of our method over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-05-21T01:47:17Z)
• Robust Regularized Low-Rank Matrix Models for Regression and Classification [14.698622796774634]
  We propose a framework for matrix variate regression models based on a rank constraint, vector regularization (e.g., sparsity), and a general loss function. We show that the algorithm is guaranteed to converge, all accumulation points of the algorithm have estimation errors in the order of $O(sqrtn)$ally and substantially attaining the minimax rate.
  arXiv  Detail & Related papers  (2022-05-14T18:03:48Z)
• Learning a Compressive Sensing Matrix with Structural Constraints via Maximum Mean Discrepancy Optimization [17.104994036477308]
  We introduce a learning-based algorithm to obtain a measurement matrix for compressive sensing related recovery problems. Recent success of such metrics in neural network related topics motivate a solution of the problem based on machine learning.
  arXiv  Detail & Related papers  (2021-10-14T08:35:54Z)
• Nonparametric Trace Regression in High Dimensions via Sign Series Representation [13.37650464374017]
  We develop a framework for nonparametric trace regression models via structured sign series representations of high dimensional functions. In the context of matrix completion, our framework leads to a substantially richer model based on what we coin as the "sign rank" of a matrix.
  arXiv  Detail & Related papers  (2021-05-04T22:20:00Z)
• Feature Weighted Non-negative Matrix Factorization [92.45013716097753]
  We propose the Feature weighted Non-negative Matrix Factorization (FNMF) in this paper. FNMF learns the weights of features adaptively according to their importances. It can be solved efficiently with the suggested optimization algorithm.
  arXiv  Detail & Related papers  (2021-03-24T21:17:17Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (or : How to Prove Kabashima's Replica Formula) [23.15629681360836]
  We prove an analytical formula for the reconstruction performance of convex generalized linear models. We show that an analytical continuation may be carried out to extend the result to convex (non-strongly) problems. We illustrate our claim with numerical examples on mainstream learning methods.
  arXiv  Detail & Related papers  (2020-06-11T16:26:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.