Finding sparse solutions of systems of polynomial equations via group-sparsity optimization
2015 (English)In: Journal of Global Optimization, ISSN 0925-5001, E-ISSN 1573-2916, Vol. 62, no 2, 319-349 p.Article in journal (Refereed) Published
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.
Place, publisher, year, edition, pages
Springer Verlag (Germany) , 2015. Vol. 62, no 2, 319-349 p.
Sparsity; Compressed sensing; Convex relaxation; Basis pursuit; Phase retrieval; Polynomial systems
IdentifiersURN: urn:nbn:se:liu:diva-119240DOI: 10.1007/s10898-014-0225-8ISI: 000354488900006OAI: oai:DiVA.org:liu-119240DiVA: diva2:821257
Funding Agencies|NSF project FORCES (Foundations Of Resilient CybEr-physical Systems); Swedish Research Council in the Linnaeus center CADICS; European Research Council under the advanced grant LEARN ; Sweden-America Foundation; Swedish Research Council2015-06-152015-06-122015-07-02