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

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

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-o...

Counting Stable Solutions of Sparse Polynomial Systems

Overdetermined Systems of Sparse Polynomial Equations

We show that, for a system of univariate polynomials given in the sparse encoding, we can compute a single polynomial de ning the same zero set, in time quasi-linear in the logarithm of the degree. In particular, it is possible to determine if such a system of polynomials does have a zero in time quasi-linear in the logarithm of the degree. The underlying algorithm relies on a result of Bombier...

Sparse phase retrieval via group-sparse optimization

This paper deals with sparse phase retrieval, i.e., the problem of estimating a vector from quadratic measurements under the assumption that few components are nonzero. In particular, we consider the problem of finding the sparsest vector consistent with the measurements and reformulate it as a group-sparse optimization problem with linear constraints. Then, we analyze the convex relaxation of ...

Finding All Solutions to a System of Polynomial Equations

Given a polynomial equation of degree d over the complex domain, the Fundamental Theorem of Algebra tells us that there are d solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard «th degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Funda...