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A Note On the Rank of the Optimal Matrix in Symmetric Toeplitz Matrix Completion Problem

Xihong Yan, Jiahao Guo, Yi Xu

TL;DR

An upper bound on the rank of the symmetric Toeplitz matrix in the completion problem based on the theorem from the trigonometric moment problem and semi-infinitely problem is provided.

Abstract

We consider the symmetric Toeplitz matrix completion problem, whose matrix under consideration possesses specific row and column structures. This problem, which has wide application in diverse areas, is well-known to be computationally NP-hard. This note provides an upper bound on the objective of minimizing the rank of the symmetric Toeplitz matrix in the completion problem based on the theorems from the trigonometric moment problem and semi-infinite problem. We prove that this upper bound is less than twice the number of linear constraints of the Toeplitz matrix completion problem. Compared with previous work in the literature, ours is one of the first efforts to investigate the bound of the objective value of the Toeplitz matrix completion problem.

A Note On the Rank of the Optimal Matrix in Symmetric Toeplitz Matrix Completion Problem

TL;DR

An upper bound on the rank of the symmetric Toeplitz matrix in the completion problem based on the theorem from the trigonometric moment problem and semi-infinitely problem is provided.

Abstract

We consider the symmetric Toeplitz matrix completion problem, whose matrix under consideration possesses specific row and column structures. This problem, which has wide application in diverse areas, is well-known to be computationally NP-hard. This note provides an upper bound on the objective of minimizing the rank of the symmetric Toeplitz matrix in the completion problem based on the theorems from the trigonometric moment problem and semi-infinite problem. We prove that this upper bound is less than twice the number of linear constraints of the Toeplitz matrix completion problem. Compared with previous work in the literature, ours is one of the first efforts to investigate the bound of the objective value of the Toeplitz matrix completion problem.

Paper Structure

This paper contains 4 sections, 3 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper Bound for the Rank of the Optimal Matrix in Symmetric Toeplitz Matrix Completion Problem
  4. Conclusion

Key Result

Lemma 1

The trigonometric moment problem has a solution, i.e., $\{\alpha_j\}_{j=0}^{n-1}$ is a valid sequence of Fourier coefficients of some positive Borel measure $\mu(t)$ defined on $[0, 2\pi]$, if and only if there exists a Hermite Toeplitz matrix $T$ that is positive semidefinite, where

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Theorem 1