The Minimum-Rank Gram Matrix Completion via Modified Fixed Point Continuation Method

TL;DR

This paper introduces algorithms based on a modified fixed point continuation method to efficiently compute minimum-rank Gram matrices, aiding in polynomial sum of squares decompositions with proven convergence.

Contribution

The paper presents novel algorithms utilizing a modified FPC approach with acceleration techniques for minimum-rank Gram matrix completion, including convergence analysis.

Findings

Algorithms effectively compute approximate and exact SOS decompositions.

The methods demonstrate convergence and efficiency in polynomial applications.

Accelerated algorithms outperform traditional approaches in speed and accuracy.

Abstract

The problem of computing a representation for a real polynomial as a sum of minimum number of squares of polynomials can be casted as finding a symmetric positive semidefinite real matrix (Gram matrix) of minimum rank subject to linear equality constraints. In this paper, we propose algorithms for solving the minimum-rank Gram matrix completion problem, and show the convergence of these algorithms. Our methods are based on the modified fixed point continuation (FPC) method. We also use the Barzilai-Borwein (BB) technique and a specific linear combination of two previous iterates to accelerate the convergence of modified FPC algorithms. We demonstrate the effectiveness of our algorithms for computing approximate and exact rational sum of squares (SOS) decompositions of polynomials with rational coefficients.