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

Abstract

The problem of verifying the nonnegativity of a real valued function on a finite set is a long-standing challenging problem, which has received extensive attention from both mathematicians and computer scientists. Given a finite set X together with a function F : X → R, if we equip X a group structure G via a bijection ϕ : G → X, then effectively verifying the nonnegativity of F on X is equivalent to computing a sparse Fourier sum of squares (FSOS) certificate of f = F • ϕ on G. In this paper, we show that by performing the fast (inverse) Fourier transform and finding a local minimal Fourier support, we are able to compute a sparse FSOS certificate of f on G with complexity O(|G| log |G| + |G|t 4 + poly(t)), which is quasi-linear in the order of G and polynomial in the FSOS sparsity t of f . We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of order up to 10 6 . It is also noticeable that different choices of group structures on X would result in different FSOS sparsities of F . Along this line, we investigate upper bounds for FSOS sparsities with respect to different choices of group structures, which generalize and refine existing results in the literature. More precisely, (i) we give an upper bound for FSOS sparsities of nonnegative functions on the product and the quotient of two finite abelian groups respectively; (ii) we prove the equivalence between finding the group structure for the Fourier-sparsest representation of F and solving an integer linear programming problem.

Short certificates for MAX-SAT via Fourier sum of squares

“…Fourier sum of squares on C n 2 . This subsection concerns with the theory of FSOS developed in [FSP16,STKI17,YYZ22]. Let f be a nonnegative function on C n 2 .…”

“…Due to its diverse applications in various fields such as the interpretability of black-box models in machine learning [BLT21], accuracy certification [NOR10] and automated reasoning [HKM16,Sha94,Par00], constructing a certificate for a computational problem is at the core of complexity theory. This paper is concerned with computation and validation of certificates for MAX-SAT, via Fourier sum of squares (FSOS) [FSP16,STKI17,YYZ22]. We formally state the two problems to be addressed in this paper as follows:…”

“…By this procedure, the problem of certifying MAX-SAT is converted to the problem of certifying the nonnegativity of functions on a group. This naturally leads to the theory of FSOS developed in [FSP16,STKI17,YYZ22]. Unfortunately, we can not directly adapt the general results and algorithms of FSOS to our particular situation.…”

“…More precisely, the degree bound of an FSOS certificate given by [STKI17, Theorem 3.2] can not ensure the existence of short certificates. Furthermore, the algorithm for FSOS in [YYZ22] requires a Fourier transform on a group, whose adaptation to our case becomes an exponential-time algorithm. To obtain better degree bounds and more efficient algorithms, we borrow tools from the approximation theory and semidefinite programming.…”

Lower Bounds of Functions on Finite Abelian Groups

“…Among those successful applications of the SDP technique, the most well-known ones are MAX-2SAT [16], MAX-3SAT [20] and MAX-CUT [22]. On the other side, it is noticed in [12,41,52,53] that if a finite set is equipped with an abelian group structure, then one can efficiently certify nonnegative functions on it by Fourier sum of squares (FSOS). Motivated by these works, this paper is concerned with establishing a framework to solve the following problem by FSOS and SDP.…”