Hanlin Zhu

Cyrus Rashtchian, David P. Woodruff, Peng Ye et al.

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.

Cyrus Rashtchian, David P. Woodruff, Hanlin Zhu

We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of $\boldsymbol{u}^{\mathrm{T}}\boldsymbol{M}\boldsymbol{v}$ over a fixed field $\mathbb{F}$ for a specified pair of vectors $\boldsymbol{u},\boldsymbol{v} \in \mathbb{F}^n$. To motivate these queries, we observe that they generalize many previously studied models, such as independent set queries, cut queries, and standard graph queries. They also specialize the recently studied matrix-vector query model. Our work is exploratory and broad, and we provide new upper and lower bounds for a wide variety of problems, spanning linear algebra, statistics, and graphs. Many of our results are nearly tight, and we use diverse techniques from linear algebra, randomized algorithms, and communication complexity.