Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recovery

• URL: http://arxiv.org/abs/2410.16826v1
• Date: Tue, 22 Oct 2024 08:58:44 GMT
• Title: Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recovery
• Authors: Paris Giampouras, HanQin Cai, Rene Vidal,
• Abstract summary: We focus on a matrix factorization-based approach for robust low-rank and asymmetric matrix recovery from corrupted measurements. We propose a subgradient algorithm that inherits the merits of preconditioned algorithms, whose rate of convergence does not depend on the condition number of the sought matrix.
• Score: 8.722715843502321
• License:
• Abstract: In this paper, we focus on a matrix factorization-based approach for robust low-rank and asymmetric matrix recovery from corrupted measurements. We address the challenging scenario where the rank of the sought matrix is unknown and employ an overparameterized approach using the variational form of the nuclear norm as a regularizer. We propose a subgradient algorithm that inherits the merits of preconditioned algorithms, whose rate of convergence does not depend on the condition number of the sought matrix, and addresses their current limitation, i.e., the lack of convergence guarantees in the case of asymmetric matrices with unknown rank. In this setting, we provide, for the first time in the literature, linear convergence guarantees for the derived overparameterized preconditioned subgradient algorithm in the presence of gross corruptions. Additionally, by applying our approach to matrix sensing, we highlight its merits when the measurement operator satisfies the mixed-norm restricted isometry properties. Lastly, we present numerical experiments that validate our theoretical results and demonstrate the effectiveness of our approach.

Related papers

• Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation [49.67011673289242]
  This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a smooth manifold. It induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure.
  arXiv  Detail & Related papers  (2023-11-08T15:17:13Z)
• A Validation Approach to Over-parameterized Matrix and Image Recovery [29.29430943998287]
  We consider the problem of recovering a low-rank matrix from several random linear measurements. We show that the proposed validation approach can also be efficiently used for image prior, an an image with a deep network.
  arXiv  Detail & Related papers  (2022-09-21T22:01:23Z)
• Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery [37.05862765171643]
  We consider a robust factorization of a low-rank matrix with no prior knowledge on the rank. We show that our particular design of diminishing the size of the matrix effectively prevents overfitting under overized models.
  arXiv  Detail & Related papers  (2021-09-23T05:54:46Z)
• Adversarially-Trained Nonnegative Matrix Factorization [77.34726150561087]
  We consider an adversarially-trained version of the nonnegative matrix factorization. In our formulation, an attacker adds an arbitrary matrix of bounded norm to the given data matrix. We design efficient algorithms inspired by adversarial training to optimize for dictionary and coefficient matrices.
  arXiv  Detail & Related papers  (2021-04-10T13:13:17Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• 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)
• Low-Rank Matrix Recovery with Scaled Subgradient Methods: Fast and Robust Convergence Without the Condition Number [34.0533596121548]
  Many problems in data science can be treated as estimating a low-rank from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are optimized via first-order methods over a smooth loss.
  arXiv  Detail & Related papers  (2020-10-26T06:21:14Z)
• A Scalable, Adaptive and Sound Nonconvex Regularizer for Low-rank Matrix Completion [60.52730146391456]
  We propose a new non scalable low-rank regularizer called "nuclear Frobenius norm" regularizer, which is adaptive and sound. It bypasses the computation of singular values and allows fast optimization by algorithms. It obtains state-of-the-art recovery performance while being the fastest in existing matrix learning methods.
  arXiv  Detail & Related papers  (2020-08-14T18:47:58Z)
• 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)
• Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion [101.83262280224729]
  We develop a relative error bound for nuclear norm regularized matrix completion. We derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix.
  arXiv  Detail & Related papers  (2015-04-26T13:12:16Z)

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.