Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets

• URL: http://arxiv.org/abs/2012.09768v2
• Date: Tue, 6 Apr 2021 15:10:50 GMT
• Title: Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets
• Authors: T. Mitchell Roddenberry, Santiago Segarra, Anastasios Kyrillidis
• Abstract summary: We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. We demonstrate practical implications by applying conic projection methods for PSD matrix recovery without incorporating low-rank regularization.
• Score: 26.42912954945887
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one projections of an approximately low-rank PSD matrix, we characterize the radius of the set of PSD matrices that satisfy the measurements. This result yields a sampling rate to guarantee singleton solution sets when the true matrix is exactly low-rank, such that the choice of the objective function or the algorithm to be used is inconsequential in its recovery. We discuss applications of this contribution and compare it to recent literature regarding implicit regularization for similar problems. We demonstrate practical implications of this result by applying conic projection methods for PSD matrix recovery without incorporating low-rank regularization.

Related papers

• Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recovery [8.722715843502321]
  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.
  arXiv  Detail & Related papers  (2024-10-22T08:58:44Z)
• Spectral Entry-wise Matrix Estimation for Low-Rank Reinforcement Learning [53.445068584013896]
  We study matrix estimation problems arising in reinforcement learning (RL) with low-rank structure. In low-rank bandits, the matrix to be recovered specifies the expected arm rewards, and for low-rank Markov Decision Processes (MDPs), it may for example characterize the transition kernel of the MDP. We show that simple spectral-based matrix estimation approaches efficiently recover the singular subspaces of the matrix and exhibit nearly-minimal entry-wise error.
  arXiv  Detail & Related papers  (2023-10-10T17:06:41Z)
• Sufficient dimension reduction for feature matrices [3.04585143845864]
  We propose a method called principal support matrix machine (PSMM) for the matrix sufficient dimension reduction. Our numerical analysis demonstrates that the PSMM outperforms existing methods and has strong interpretability in real data applications.
  arXiv  Detail & Related papers  (2023-03-07T23:16:46Z)
• 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)
• Non-PSD Matrix Sketching with Applications to Regression and Optimization [56.730993511802865]
  We present dimensionality reduction methods for non-PSD and square-roots" matrices. We show how these techniques can be used for multiple downstream tasks.
  arXiv  Detail & Related papers  (2021-06-16T04:07:48Z)
• Solving weakly supervised regression problem using low-rank manifold regularization [77.34726150561087]
  We solve a weakly supervised regression problem. Under "weakly" we understand that for some training points the labels are known, for some unknown, and for others uncertain due to the presence of random noise or other reasons such as lack of resources. In the numerical section, we applied the suggested method to artificial and real datasets using Monte-Carlo modeling.
  arXiv  Detail & Related papers  (2021-04-13T23:21:01Z)
• Robust Low-rank Matrix Completion via an Alternating Manifold Proximal Gradient Continuation Method [47.80060761046752]
  Robust low-rank matrix completion (RMC) has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims to decompose a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) and the l1 norm of the sparse matrix (to promote sparsity). In this paper, motivated by some recent works on low-
  arXiv  Detail & Related papers  (2020-08-18T04:46:22Z)
• Positive Semidefinite Matrix Factorization: A Connection with Phase Retrieval and Affine Rank Minimization [71.57324258813674]
  We show that PSDMF algorithms can be designed based on phase retrieval (PR) and affine rank minimization (ARM) algorithms. Motivated by this idea, we introduce a new family of PSDMF algorithms based on iterative hard thresholding (IHT)
  arXiv  Detail & Related papers  (2020-07-24T06:10:19Z)
• Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform [64.90148466525754]
  We study the performance of iterative sketching for least-squares problems. We show that the convergence rate for Haar and randomized Hadamard matrices are identical, andally improve upon random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.
  arXiv  Detail & Related papers  (2020-02-03T16:17:50Z)

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.