Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic representation theory of finite groups indicates that a function f on a finite abelian group G can be written as a linear combination of characters of irreducible representations of G by $f\left(x\right)={\sum }_{\chi \in \hat{G}}\hat{f}\left(\chi \right)\chi \left(x\right)$, where $\hat{G}$ is the dual group of G consisting of all characters of G and $\hat{f}\left(\chi \right)$ is the Fourier coefficient of f at $\chi \in \hat{G}$. In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of f on a finite abelian group G with complexity that is quasi-linear in the order of G and polynomial in the FSOS sparsity of f. Moreover, for a nonnegative function f on a finite abelian group G and a subset $S\subseteq \hat{G}$, we give a lower bound of the constant M such that $f+M$ admits an FSOS supported on S. We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to 10⁷. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem by sparse FSOS.