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 theory of 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 $f(x)=\sum_{\chi \in \widehat{G}} \widehat{f} (\chi)\chi(x)$, where $\widehat{G}$ is the dual group of $G$ consisting of all characters of $G$ and $\widehat{f} (\chi)$ is the Fourier coefficient of $f$ at $\chi \in \widehat{G}$. 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 \if $\operatorname{O}\left(|G| \log(|G|)+\log(k_{\min})\operatorname{SDP}(2k_{\min})\right)$,\fi that is quasi-linear in the order of $G$ and polynomial in the FSOS sparsity \if $k_{\min}$\fi of $f$. Moreover, for a nonnegatvie function $f$ on a finite abelian group $G$ and a set $S \subset \widehat{G}$, 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 $10^7$. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem \if on $\mathbb{T}^n$\fi by sparse FSOS.