Mathematical Programming

, Volume 160, Issue 1–2, pp 149–191

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

• Hamza Fawzi
• James Saunderson
• Pablo A. Parrilo
Full Length Paper Series A

## 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 dth 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

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## Authors and Affiliations

• Hamza Fawzi
• 1
• James Saunderson
• 2
• Pablo A. Parrilo
• 1
1. 1.Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA
2. 2.Department of Electrical EngineeringUniversity of WashingtonSeattleUSA