Average-Case Communication Complexity of Statistical Problems

TL;DR

This paper investigates the average-case communication complexity of statistical problems like planted clique and sparse PCA, providing new lower bounds and a reduction method that preserves input distributions in random graph or matrix models.

Contribution

It introduces a general reduction technique for average-case problems and derives new communication lower bounds for various statistical detection tasks.

Findings

Established nearly tight lower bounds for planted clique detection.

Derived new bounds for query complexity in multiple sketching models.

Provided simplified proofs for existing lower bounds in the edge-probe model.

Abstract

We study statistical problems, such as planted clique, its variants, and sparse principal component analysis in the context of average-case communication complexity. Our motivation is to understand the statistical-computational trade-offs in streaming, sketching, and query-based models. Communication complexity is the main tool for proving lower bounds in these models, yet many prior results do not hold in an average-case setting. We provide a general reduction method that preserves the input distribution for problems involving a random graph or matrix with planted structure. Then, we derive two-party and multi-party communication lower bounds for detecting or finding planted cliques, bipartite cliques, and related problems. As a consequence, we obtain new bounds on the query complexity in the edge-probe, vector-matrix-vector, matrix-vector, linear sketching, and $\mathbb{F}_2$-sketching models. Many of these results are nearly tight, and we use our techniques to provide simple proofs of some known lower bounds for the edge-probe model.