Reducibility and Computational Lower Bounds for Problems with Planted Sparse Structure

TL;DR

This paper establishes strong computational lower bounds for various high-dimensional statistical problems with planted sparse structures, using novel average-case reductions based on the planted clique conjecture, and matches these bounds with algorithms.

Contribution

Introduces new techniques for average-case reductions to prove computational lower bounds for problems with planted sparse structures, connecting them to the planted clique conjecture.

Findings

Established tight lower bounds for multiple planted problems

Developed algorithms that match the lower bounds

Identified the information-theoretic limits of the models

Abstract

The prototypical high-dimensional statistics problem entails finding a structured signal in noise. Many of these problems exhibit an intriguing phenomenon: the amount of data needed by all known computationally efficient algorithms far exceeds what is needed for inefficient algorithms that search over all possible structures. A line of work initiated by Berthet and Rigollet in 2013 has aimed to explain these statistical-computational gaps by reducing from conjecturally hard average-case problems in computer science. However, the delicate nature of average-case reductions has limited the applicability of this approach. In this work we introduce several new techniques to give a web of average-case reductions showing strong computational lower bounds based on the planted clique conjecture using natural problems as intermediates. These include tight lower bounds for Planted Independent Set, Planted Dense Subgraph, Sparse Spiked Wigner, Sparse PCA, a subgraph variant of the Stochastic Block Model and a biased variant of Sparse PCA. We also give algorithms matching our lower bounds and identify the information-theoretic limits of the models we consider.