  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.
