Toeplitz matrix completion via a low-rank approximation algorithm

Ruiping Wen,Yaru Fu

doi:10.1186/s13660-020-02340-w

Abstract

In this paper, we propose a low-rank matrix approximation algorithm for solving the Toeplitz matrix completion (TMC) problem. The approximation matrix was obtained by the mean projection operator on the set of feasible Toeplitz matrices for every iteration step. Thus, the sequence of the feasible Toeplitz matrices generated by iteration is of Toeplitz structure throughout the process, which reduces the computational time of the singular value decomposition (SVD) and approximates well the solution. On the theoretical side, we provide a convergence analysis to show that the matrix sequences of iterates converge. On the practical side, we report the numerical results to show that the new algorithm is more effective than the other algorithms for the TMC problem.

Highlights

The problem of completing a low-rank matrix, which is to recover a matrix from the partial entries, has became more and more popular
It is well known that a manifold of rank r matrix can be factorized into a bi-linear form: X = GHT with G ∈ Rm×r, H ∈ Rn×r, so some algorithms have been proposed to solve the problem (1.1) or (1.2), for example, Vandereycken [27] proposed the geometric conjugate gradient method and Newton method on Riemannian manifold; Mishra [21], Boumal and Absil [5] presented the Riemannian geometry method, i.e., the gradient descent scheme and trust-region scheme; Tanner and Wei [26] took advantage of the different descent directions for exact linear search respectively; the alternating direction methods were presented by Jain [15]; Keshavan [16] and Wen [38] presented gradient descent method and nonlinear successive over-relaxation algorithm respectively
There are a lot algorithms which are effective for the low-rank Hankel or Toeplitz matrix completion problem

Summary

Introduction

The problem of completing a low-rank matrix, which is to recover a matrix from the partial entries, has became more and more popular. There are a lot algorithms which are effective for the low-rank Hankel or Toeplitz matrix completion problem. By utilizing the low-rank structure of the Hankel matrix corresponding to a spectrally sparse signal x, [8] introduced a computationally efficient algorithm for the spectral compressed sensing problem.

Results

Conclusion