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

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 × n PSD matrices, denoted by D , using at most N number of 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 × n symmetric matrices that are PSD when restricted to a fixed set of k -dimensional subspaces in 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) . Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial S_+^k -extension complexity. To be more precise, if S is a constant factor approximation of D , then S must have S_+^k -extension complexity at least exp ( C ·min{√(n), n/k }) where C is some absolute constant. In addition, we show that any set S such that D ⊆ S and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have S_+^k -extension complexity at least exp ( C ·min{ n^1/3, √(n/k)}) . These results imply that the cone of n × n PSD matrices cannot be approximated by a polynomial number of k × k PSD constraints for any 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 S_+^n , which corresponds to the special case with k=1 .