A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

8 years 1 months ago

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Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix M, the goal is to compute a matrix M of given rank r in a linear or aﬃne subspace E of matrices (usually encoding a speciﬁc structure) such that the Frobenius distance M − M is small. We propose a Newton-like iteration for solving this problem, whose main feature is that it converges locally quadratically to such a matrix under mild transversality assumptions between the manifold of matrices of rank r and the linear/aﬃne subspace E. We also show that the distance between the limit of the iteration and the optimal solution of the problem is quadratic in the distance between the input matrix and the manifold of rank r matrices in E. To illustrate the applicability of this algorithm, we propose a Maple implementation and give experimental results for several applicative problems that can be modeled by Structured Low-Rank Appr...

Éric Schost, Pierre-Jean Spaenlehauer

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