Kronecker Product Approximation of Operators in Spectral Norm via Alternating SDP

TL;DR

This paper introduces an alternating SDP-based method for approximating linear operators with Kronecker products in spectral norm, addressing a challenging problem with practical computational benefits.

Contribution

It proposes a novel alternating optimization approach using semidefinite programming for spectral norm Kronecker approximation, improving over traditional Frobenius norm solutions.

Findings

The method achieves high-quality spectral norm approximations.

Computational experiments demonstrate the effectiveness of the approach.

The approach outperforms existing methods in spectral norm approximation.

Abstract

The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The approximation problem in Frobenius norm admits a well-known solution via the singular value decomposition. However, the approximation problem in spectral norm, which is more natural for linear operators, is much more challenging. In particular, the Frobenius norm solution can be far from optimal in spectral norm. We describe an alternating optimization method based on semidefinite programming to obtain high-quality approximations in spectral norm, and we present computational experiments to illustrate the advantages of our approach.