Non-convex matrix sensing: Breaking the quadratic rank barrier in the sample complexity

• URL: http://arxiv.org/abs/2408.13276v2
• Date: Tue, 10 Sep 2024 18:44:10 GMT
• Title: Non-convex matrix sensing: Breaking the quadratic rank barrier in the sample complexity
• Authors: Dominik Stöger, Yizhe Zhu,
• Abstract summary: We study the problem of reconstructing a low-rank quadratic convex matrix from a few measurements. We show that factorized gradient with spectral specificity converges to truth with the number of samples.
• Score: 11.412228884390784
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: For the problem of reconstructing a low-rank matrix from a few linear measurements, two classes of algorithms have been widely studied in the literature: convex approaches based on nuclear norm minimization, and non-convex approaches that use factorized gradient descent. Under certain statistical model assumptions, it is known that nuclear norm minimization recovers the ground truth as soon as the number of samples scales linearly with the number of degrees of freedom of the ground-truth. In contrast, while non-convex approaches are computationally less expensive, existing recovery guarantees assume that the number of samples scales at least quadratically with the rank $r$ of the ground-truth matrix. In this paper, we close this gap by showing that the non-convex approaches can be as efficient as nuclear norm minimization in terms of sample complexity. Namely, we consider the problem of reconstructing a positive semidefinite matrix from a few Gaussian measurements. We show that factorized gradient descent with spectral initialization converges to the ground truth with a linear rate as soon as the number of samples scales with $ \Omega (rd\kappa^2)$, where $d$ is the dimension, and $\kappa$ is the condition number of the ground truth matrix. This improves the previous rank-dependence in the sample complexity of non-convex matrix factorization from quadratic to linear. Our proof relies on a probabilistic decoupling argument, where we show that the gradient descent iterates are only weakly dependent on the individual entries of the measurement matrices. We expect that our proof technique is of independent interest for other non-convex problems.

Related papers

• A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval [56.67706781191521]
  In this work, we present a robust phase retrieval problem where the task is to recover an unknown signal. Our proposed oracle avoids the need for computationally spectral descent, using a simple gradient step and outliers.
  arXiv  Detail & Related papers  (2024-09-07T06:37:23Z)
• Entrywise error bounds for low-rank approximations of kernel matrices [55.524284152242096]
  We derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues. We validate our theory with an empirical study of a collection of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2024-05-23T12:26:25Z)
• The Inductive Bias of Flatness Regularization for Deep Matrix Factorization [58.851514333119255]
  This work takes the first step toward understanding the inductive bias of the minimum trace of the Hessian solutions in deep linear networks. We show that for all depth greater than one, with the standard Isometry Property (RIP) on the measurements, minimizing the trace of Hessian is approximately equivalent to minimizing the Schatten 1-norm of the corresponding end-to-end matrix parameters.
  arXiv  Detail & Related papers  (2023-06-22T23:14:57Z)
• Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squares [0.8702432681310401]
  We propose a new algorithm for recovering data that adheres to multiple, heterogeneous low-dimensional structures from linear observations. We show that the IRLS method favorable in identifying low/comckuele state measurements.
  arXiv  Detail & Related papers  (2023-06-08T06:35:47Z)
• Distributional Hardness Against Preconditioned Lasso via Erasure-Robust Designs [22.41443027099101]
  We show that standard sparse random designs are with high probability robust to adversarial measurement erasures. This is the first time that partial recoverability of arbitrary sparse signals under erasures has been studied in compressed sensing.
  arXiv  Detail & Related papers  (2022-03-05T22:16:05Z)
• Stochastic regularized majorization-minimization with weakly convex and multi-convex surrogates [0.0]
  We show that the first optimality gap of the proposed algorithm decays at the rate of the expected loss for various methods under nontens dependent data setting. We obtain first convergence point for various methods under nontens dependent data setting.
  arXiv  Detail & Related papers  (2022-01-05T15:17:35Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• Beyond Procrustes: Balancing-Free Gradient Descent for Asymmetric Low-Rank Matrix Sensing [36.96922859748537]
  Low-rank matrix estimation plays a central role in various applications across science and engineering. Existing approaches rely on adding a metric regularization term to balance the scale of the two matrix factors. In this paper, we provide a theoretical justification for the performance in recovering a low-rank matrix from a small number of linear measurements.
  arXiv  Detail & Related papers  (2021-01-13T15:03:52Z)
• Linear-Sample Learning of Low-Rank Distributions [56.59844655107251]
  We show that learning $ktimes k$, rank-$r$, matrices to normalized $L_1$ distance requires $Omega(frackrepsilon2)$ samples. We propose an algorithm that uses $cal O(frackrepsilon2log2fracepsilon)$ samples, a number linear in the high dimension, and nearly linear in the matrices, typically low, rank proofs.
  arXiv  Detail & Related papers  (2020-09-30T19:10:32Z)
• Learning Mixtures of Low-Rank Models [89.39877968115833]
  We study the problem of learning computational mixtures of low-rank models. We develop an algorithm that is guaranteed to recover the unknown matrices with near-optimal sample. In addition, the proposed algorithm is provably stable against random noise.
  arXiv  Detail & Related papers  (2020-09-23T17:53:48Z)
• Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent [34.0533596121548]
  Low-rank matrix estimation converges convex problem that finds numerous applications in signal processing, machine learning and imaging science. We show that ScaledGD achieves the best of the best in terms of the number of the low-rank matrix. Our analysis is also applicable to general loss that are similar to low-rank gradient descent.
  arXiv  Detail & Related papers  (2020-05-18T17:17:16Z)

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.