On approximations of the PSD cone by a polynomial number of smaller-sized PSD cones

Dogyoon Song, Pablo A. Parrilo

Abstract: We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: “how closely can we approximate the set of unit-trace $$n \times n$$ n × n PSD matrices, denoted by D, using at most N number of $$k \times k$$ k × k PSD constraints?” In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace $$n \times n$$ n × n symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in $${\mathbb {R}}^n$$ R n . We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any $$k = o(n)$$ k = o ( n ) . Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial $${\varvec{S}}_+^k$$ S + k -extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have $${\varvec{S}}_+^k$$ S + k -extension complexity at least $$\exp ( C \cdot \min \{ \sqrt{n}, n/k \})$$ exp ( C · min { n , n / k } ) where C is some absolute constant. In addition, we show that any set S such that $$D \subseteq S$$ D ⊆ S and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have $${\varvec{S}}_+^k$$ S + k -extension complexity at least $$\exp ( C \cdot \min \{ n^{1/3}, \sqrt{n/k} \})$$ exp ( C · min { n 1 / 3 , n / k } ) . These results imply that the cone of $$n \times n$$ n × n PSD matrices cannot be approximated by a polynomial number of $$k \times k$$ k × k PSD constraints for any $$k = o(n / \log ^2 n)$$ k = o ( n / log 2 n ) . These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of $${\varvec{S}}_+^n$$ S + n , which corresponds to the special case with $$k=1$$ k = 1 .