Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems

We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of u^TMv over a fixed field 𝔽 for a specified pair of vectors u,v∈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.