Finding sparse solutions of systems of polynomial equations via group-sparsity optimization
This addresses a fundamental computational challenge in fields like signal processing and optimization by providing efficient algorithms for sparse polynomial systems, though it is incremental as it builds on existing group-sparsity techniques.
The paper tackles the problem of finding sparse solutions to systems of polynomial equations, even with noise, by converting it into a group-sparsity optimization problem and proposing two methods: a convex relaxation with exact recovery guarantees and a greedy algorithm for efficiency, achieving high success probabilities and fast computing times compared to prior work.
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.