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

TL;DR

This paper develops a general framework for establishing Sum-of-Squares lower bounds on various certification problems, extending previous techniques to new models like tensor PCA and sparse PCA.

Contribution

It introduces a unified machinery that generalizes prior methods for proving Sum-of-Squares lower bounds across multiple problems.

Findings

Proves degree $n^{ ext{ extbackslash epsilon}}$ Sum-of-Squares lower bounds for tensor PCA.

Establishes similar bounds for the Wishart model of sparse PCA.

Provides lower bounds for a variant of planted clique called planted slightly denser subgraph.

Abstract

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.