# Sparse sums of squares on finite abelian groups and improved semidefinite lifts

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## Abstract

Let *G* be a finite abelian group. This paper is concerned with nonnegative functions on *G* that are *sparse* with respect to the Fourier basis. We establish combinatorial conditions on subsets \({\mathcal {S}}\) and \({\mathcal {T}}\) of Fourier basis elements under which nonnegative functions with Fourier support \({\mathcal {S}}\) are sums of squares of functions with Fourier support \({\mathcal {T}}\). Our combinatorial condition involves constructing a chordal cover of a graph related to *G* and \({\mathcal {S}}\) (the Cayley graph \({{\mathrm{Cay}}}(\widehat{G},{\mathcal {S}})\)) with maximal cliques related to \({\mathcal {T}}\). Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of *G* (the characters of *G*). We apply our general result to two examples. First, in the case where \(G = {\mathbb {Z}}_2^n\), by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in *n* binary variables is a sum of squares of functions of degree at most \(\left\lceil n/2 \right\rceil \), establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree *d* on \({\mathbb {Z}}_N\) (when *d* divides *N*). By constructing a particular chordal cover of the *d*th power of the *N*-cycle, we prove that any such function is a sum of squares of functions with at most \(3d\log (N/d)\) nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in \({\mathbb {R}}^{2d}\) with *N* vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size \(3d\log (N/d)\). Putting \(N=d^2\) gives a family of polytopes in \({\mathbb {R}}^{2d}\) with linear programming extension complexity \(\varOmega (d^2)\) and semidefinite programming extension complexity \(O(d\log (d))\). To the best of our knowledge, this is the first explicit family of polytopes \((P_d)\) in increasing dimensions where \({{\mathrm{xc_{\text {PSD}}}}}(P_d) = o({{\mathrm{xc_{\text {LP}}}}}(P_d))\), where \({{\mathrm{xc_{\text {PSD}}}}}\) and \({{\mathrm{xc_{\text {LP}}}}}\) are respectively the SDP and LP extension complexity.

## Mathematics Subject Classification

90C22 52B12 52B55## Supplementary material

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