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

- 405 Downloads
- 5 Citations

## 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

## References

- 1.Agler, J., Helton, W., McCullough, S., Rodman, L.: Positive semidefinite matrices with a given sparsity pattern. Linear Algebra Appl.
**107**, 101–149 (1988)MathSciNetCrossRefMATHGoogle Scholar - 2.Barvinok, A.: A Course in Convexity, vol. 54. American Mathematical Society, Providence (2002)MATHGoogle Scholar
- 3.Bogomolov, Y., Fiorini, S., Maksimenko, A., Pashkovich, K.: Small extended formulations for cyclic polytopes. Discrete Comput Geom.
**53**(4), 809–816 (2015). doi: 10.1007/s00454-015-9682-1 - 4.Blekherman, G., Gouveia, J., Pfeiffer, J.: Sums of squares on the hypercube. arXiv preprint arXiv:1402.4199 (2014)
- 5.Blekherman, G., Parrilo, P.A., Thomas, R.R.: Semidefinite optimization and convex algebraic geometry. SIAM (2013)Google Scholar
- 6.Dumitrescu, B.: Positive Trigonometric Polynomials and Signal Processing Applications. Springer, Berlin (2007)MATHGoogle Scholar
- 7.Fawzi, H., Gouveia, J., Parrilo, P.A., Robinson, R.Z., Thomas, R.R.: Positive semidefinite rank. Math. Program.
**153**(1), 133–177 (2015)MathSciNetCrossRefMATHGoogle Scholar - 8.Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discrete Math.
**313**(1), 67–83 (2013)MathSciNetCrossRefMATHGoogle Scholar - 9.Fawzi, H., Saunderson, J., Parrilo, P.A.: Equivariant semidefinite lifts of regular polygons. arXiv preprint arXiv:1409.4379 (2014)
- 10.Fawzi, H., Saunderson, J., Parrilo, P.A.: Equivariant semidefinite lifts and sum-of-squares hierarchies. SIAM J. Optim.
**25**(4), 2212–2243 (2015)MathSciNetCrossRefMATHGoogle Scholar - 11.Gale, D.: Neighborly and cyclic polytopes. In: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, vol. 7, pp. 225–232 (1963)Google Scholar
- 12.Grone, R., Johnson, C.R., Sá, E.M., Wolkowicz, H.: Positive definite completions of partial hermitian matrices. Linear Algebra Appl.
**58**, 109–124 (1984)MathSciNetCrossRefMATHGoogle Scholar - 13.Gouveia, J., Parrilo, P.A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Oper. Res.
**38**(2), 248–264 (2013)MathSciNetCrossRefMATHGoogle Scholar - 14.Griewank, A., Toint, P.L.: On the existence of convex decompositions of partially separable functions. Math. Program.
**28**(1), 25–49 (1984)MathSciNetCrossRefMATHGoogle Scholar - 15.Kaibel, V., Pashkovich, K.: Constructing extended formulations from reflection relations. In: Integer programming and combinatorial optimization, pp. 287–300. Springer, Berlin (2011)Google Scholar
- 16.Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim.
**11**(3), 796–817 (2001)MathSciNetCrossRefMATHGoogle Scholar - 17.Laurent, M.: Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope. Math. Oper. Res.
**28**(4), 871–883 (2003)MathSciNetCrossRefMATHGoogle Scholar - 18.Nesterov, Y.: Squared functional systems and optimization problems. In: High performance optimization, pp. 405–440. Springer, Berlin (2000)Google Scholar
- 19.Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology (2000)Google Scholar
- 20.Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1990)CrossRefMATHGoogle Scholar
- 21.Terras, A.: Fourier Analysis on Finite Groups and Applications. Cambridge University Press, Cambridge (1999)CrossRefMATHGoogle Scholar
- 22.Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci.
**43**(3), 441–466 (1991)MathSciNetCrossRefMATHGoogle Scholar - 23.Ziegler, G.M.: Lectures on Polytopes, vol. 152. Springer, Berlin (1995)MATHGoogle Scholar