Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An efficient asymptotic DC method for sparse and low-rank matrix recovery

Mingcai Ding, Xiaoliang Song, Bo Yu

TL;DR

Numerical experiments on sparse phase retrieval demonstrate that ADC-siDCA is a valuable tool for recovering sparse and low-rank Hermitian matrices, and indicates that ADC-siDCA surpasses the other two methods in terms of efficiency and recovery error.

Abstract

The optimization problem of sparse and low-rank matrix recovery is considered, which involves a least squares problem with a rank constraint and a cardinality constraint. To overcome the challenges posed by these constraints, an asymptotic difference-of-convex (ADC) method that employs a Moreau smoothing approach and an exact penalty approach is proposed to transform this problem into a DC programming format gradually. To solve the gained DC programming, by making full use of its DC structure, an efficient inexact DC algorithm with sieving strategy (siDCA) is introduced. The subproblem of siDCA is solved by an efficient dual-based semismooth Newton method. The convergence of the solution sequence generated by siDCA is proved. To illustrate the effectiveness of ADC-siDCA, matrix recovery experiments on nonnegative and positive semidefinite matrices. The numerical results are compared with those obtained using a successive DC approximation minimization method and a penalty proximal alternating linearized minimization approach. The outcome of the comparison indicates that ADC-siDCA surpasses the other two methods in terms of efficiency and recovery error. Additionally, numerical experiments on sparse phase retrieval demonstrate that ADC-siDCA is a valuable tool for recovering sparse and low-rank Hermitian matrices.

An efficient asymptotic DC method for sparse and low-rank matrix recovery

TL;DR

Numerical experiments on sparse phase retrieval demonstrate that ADC-siDCA is a valuable tool for recovering sparse and low-rank Hermitian matrices, and indicates that ADC-siDCA surpasses the other two methods in terms of efficiency and recovery error.

Abstract

The optimization problem of sparse and low-rank matrix recovery is considered, which involves a least squares problem with a rank constraint and a cardinality constraint. To overcome the challenges posed by these constraints, an asymptotic difference-of-convex (ADC) method that employs a Moreau smoothing approach and an exact penalty approach is proposed to transform this problem into a DC programming format gradually. To solve the gained DC programming, by making full use of its DC structure, an efficient inexact DC algorithm with sieving strategy (siDCA) is introduced. The subproblem of siDCA is solved by an efficient dual-based semismooth Newton method. The convergence of the solution sequence generated by siDCA is proved. To illustrate the effectiveness of ADC-siDCA, matrix recovery experiments on nonnegative and positive semidefinite matrices. The numerical results are compared with those obtained using a successive DC approximation minimization method and a penalty proximal alternating linearized minimization approach. The outcome of the comparison indicates that ADC-siDCA surpasses the other two methods in terms of efficiency and recovery error. Additionally, numerical experiments on sparse phase retrieval demonstrate that ADC-siDCA is a valuable tool for recovering sparse and low-rank Hermitian matrices.
Paper Structure (20 sections, 14 theorems, 174 equations, 7 figures, 7 algorithms)

This paper contains 20 sections, 14 theorems, 174 equations, 7 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. An asymptotic DC method for sparse and low-rank matrix recovery
  4. Exact penalty property
  5. An asymptotic DC method
  6. Inexact DCA with sieving strategy
  7. An efficient iDCA
  8. A dual-based semismooth Newton method
  9. Convergence analysis of siDCA
  10. Finite serious steps are performed in siDCA
  11. Infinite serious steps are performed in siDCA
  12. Numerical experiments
  13. Sparse and low-rank nonnegative matrix recovery
  14. Sparse and low-rank positive semidefinite matrix recovery
  15. Sparse phase retrieval
  16. ...and 5 more sections

Key Result

Proposition 1

Let $D_{\mu, g}(\mathbf{U})$ be defined in eq12, then for any $\mu>0$,

Figures (7)

  • Figure 1: Comparison of the performance for sparse phase retrieval
  • Figure 2: Comparison of sparse and low-rank matrix recovery performance of nonnegative Cliq model
  • Figure 3: Comparison of sparse and low-rank matrix recovery performance of Rand1 model
  • Figure 4: Comparison of sparse and low-rank matrix recovery performance of Rand2 model
  • Figure 5: Comparison of sparse and low-rank recovery performance of positive semidefinite clique model
  • ...and 2 more figures

Theorems & Definitions (26)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Proposition 3
  • proof
  • Theorem 4
  • Theorem 5
  • Remark 2
  • Proposition 6
  • ...and 16 more