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

Understand

In this paper, we construct general machinery for proving Sum-of-Squares lower bounds on certification problems by generalizing the techniques used by Barak et al.

• [FOCS 2016] to prove Sum-of-Squares lower bounds for planted clique.
• Using this machinery, we prove degree $n^{\epsilon}$ 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.