Machinery for Proving Sum-of-Squares Lower Bounds on Certification Problems

Abstract

In this paper, we construct general machinery for proving Sum-of-Squares lower bounds on certification problems by generalizing the techniques used by [BHK + 16] to prove Sum-of-Squares lower bounds for planted clique. Using this machinery, we prove degree n ε Sum-of-Squares lower bounds for tensor PCA, the Wishart model of sparse PCA, and a variant of planted clique which we call planted slightly denser subgraph.

Sum-of-Squares Lower Bounds for Sparse Independent Set

“…In the case of CSPs, one fixes the structure of the lower bound instance and only considers an instance to be specified by the signs of the literals, which can again be taken as uniformly random {−1, 1} variables. Similarly, recent results by a subset of the authors [PR20] for tensor PCA apply when the input tensor has independent Rademacher or Gaussian entries. The techniques used to establish these lower bounds have proved difficult to extend to the case when the input distribution naturally corresponds to a sparse graph (or more generally, when it is specified by a collection of independent sub-gaussian variables, with Orlicz norm ω(1) instead of O(1)).…”

“…Examples include the planted clique lower bound of Barak et al [BHK + 16], CSP lower bounds of Kothari et al [KMOW17], and the tensor PCA lower bounds [HKP + 17, PR20]. The technical component of decomposing the moment matrix in the dense case, as a sum of PSD matrices, is developed into a general "machinery" in a recent work by a subset of the authors [PR20]. A different approach than the ones based on pseudocalibration, which also applies in the dense regime, was developed by Kunisky [Kun20].…”

“…Norm bounds for sparse graph matrices are obtained in Section 5. Section 6 then proves the PSD-ness of our SoS solutions broadly using the machinery developed in [BHK + 16] and [PR20].…”

“…The base of the first term (vertex decay) is less than 1 for k n/ √ d. The second term has a nonnegative exponent by the intersection tradeoff lemma from [BHK + 16]. The version we cite is Lemma 9.32 in [PR20], with the simplification that τ is a proper middle shape, so…”