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

Authors Cyrus Rashtchian, David P. Woodruff, Hanlin Zhu



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Author Details

Cyrus Rashtchian
  • Department of Computer Science & Engineering, UC San Diego, CA, USA
David P. Woodruff
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Hanlin Zhu
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

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Cyrus Rashtchian, David P. Woodruff, and Hanlin Zhu. Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 26:1-26:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.26

Abstract

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

Subject Classification

ACM Subject Classification
  • Theory of computation → Stochastic approximation
Keywords
  • Query complexity
  • property testing
  • vector-matrix-vector
  • linear algebra
  • statistics
  • graph parameter estimation

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