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

## TL;DR

This paper establishes an upper bound on the rank of the optimal symmetric Toeplitz matrix in the completion problem, linking it to the number of constraints and leveraging trigonometric moment theory.

## Contribution

It provides one of the first bounds on the objective value for symmetric Toeplitz matrix completion, connecting it to classical mathematical theorems.

## Key findings

- Upper bound on the rank is less than twice the number of constraints.
- The bound is derived using trigonometric moment problem and semi-infinite problem theorems.
- This work advances understanding of the matrix completion problem's complexity.

## 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.