Finding sparse solutions of systems of polynomial equations via group-sparsity optimization

TL;DR

This paper introduces two efficient methods for recovering sparse solutions of polynomial systems, leveraging group-sparsity optimization, with theoretical guarantees in noiseless cases and empirical success in noisy scenarios.

Contribution

It proposes convex and greedy algorithms for sparse polynomial solutions, providing theoretical recovery conditions and demonstrating computational efficiency and accuracy improvements.

Findings

Convex relaxation method achieves exact recovery in noiseless cases.

Greedy approach offers computational efficiency with comparable accuracy.

Empirical analysis shows success probability relates to solution sparsity.

Abstract

The paper deals with the problem of finding sparse solutions to systems of polynomial equations possibly perturbed by noise. In particular, we show how these solutions can be recovered from group-sparse solutions of a derived system of linear equations. Then, two approaches are considered to find these group-sparse solutions. The first one is based on a convex relaxation resulting in a second-order cone programming formulation which can benefit from efficient reweighting techniques for sparsity enhancement. For this approach, sufficient conditions for the exact recovery of the sparsest solution to the polynomial system are derived in the noiseless setting, while stable recovery results are obtained for the noisy case. Though lacking a similar analysis, the second approach provides a more computationally efficient algorithm based on a greedy strategy adding the groups one-by-one. With respect to previous work, the proposed methods recover the sparsest solution in a very short computing time while remaining at least as accurate in terms of the probability of success. This probability is empirically analyzed to emphasize the relationship between the ability of the methods to solve the polynomial system and the sparsity of the solution.