Structured Low-Rank Approximation: Optimization on Matrix Manifold Approach

Abstract

We deal with the problem to compute the nearest Structured Low-Rank Approximation (SLRA) to a given matrix in this paper. This problem arises in many practical applications, such as computation of approximate GCD of polynomials, matrix completion problems, image processing and control theory etc. We reformulate this problem as an unconstrained optimization problem on an appropriately chosen Stiefel manifold. This proposed formulation is based on the condition that the computed nearest SLRA is required to have a suitable number of linearly independent vectors in its nullspace. Thus, this formulation helps in computing the nearest SLRA with exact required rank as opposed to many numerical techniques available in literature. Two different techniques to solve the optimization problem on the Stiefel manifold are discussed in this paper with several numerical case studies. The effectiveness of the proposed approach is demonstrated through numerical examples and by comparison with the benchmark methods available in the literature.