The minimum-rank gram matrix completion via modified fixed point continuation method

Authors:
Yue Ma;Lihong Zhi
Affiliations:
Key Laboratory of Mathematics Mechanization, AMSS, Beijing, China;Key Laboratory of Mathematics Mechanization, AMSS, Beijing, China
Venue:
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Year:
2011

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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 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 fixed point continuation method. We also use the Barzilai-Borwein technique and a specific linear combination of two previous iterates to accelerate the convergence of modified fixed point continuation algorithms. We demonstrate the effectiveness of our algorithms for computing approximate and exact rational sum of squares decompositions of polynomials with rational coefficients.