Variable projection for affinely structured low-rank approximation in weighted 2-norms

TL;DR

This paper introduces a variable projection approach for structured low-rank approximation in weighted 2-norms, providing efficient algorithms for mosaic Hankel matrices with reduced computational complexity.

Contribution

It develops algorithms using variable projection for affine structured low-rank approximation, including cost, gradient, and Hessian evaluations, optimized for mosaic Hankel matrices.

Findings

Algorithms achieve $O(m^2 n)$ complexity for mosaic Hankel matrices.

Efficient evaluation of cost function, gradient, and Hessian.

Applicable to general affine structures with weighted norms.

Abstract

The structured low-rank approximation problem for general affine structures, weighted 2-norms and fixed elements is considered. The variable projection principle is used to reduce the dimensionality of the optimization problem. Algorithms for evaluation of the cost function, the gradient and an approximation of the Hessian are developed. For $m \times n$ mosaic Hankel matrices the algorithms have complexity $O(m^2 n)$.