Affiliation:
1. Institute of Computing Technology, Chinese Academy of Sciences, China and University of Chinese Academy of Sciences, Beijing, China
2. Carnegie Mellon University, Pennsylvania, USA
3. Institute of Computing Technology, Chinese Academy of Sciences, China and Conflux, Beijing, China
Abstract
We consider algorithms with access to an unknown matrix M ε F
n×d
via
matrix-vector products
, namely, the algorithm chooses vectors v
1
, ⃛ , v
q
, and observes Mv
1
, ⃛ , Mv
q
. Here the v
i
can be randomized as well as chosen adaptively as a function of Mv
1
, ⃛ , Mv
i-1
. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number
q
of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph.
Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.
Funder
National Natural Science Foundation of China
Strategic Priority Research Program of Chinese Academy of Sciences
National Science Foundation
Chinese Academy of Sciences and the Simons Institute for the Theory of Computing
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Structured matrix recovery from matrix‐vector products;Numerical Linear Algebra with Applications;2023-09-22
2. Optimal Eigenvalue Approximation via Sketching;Proceedings of the 55th Annual ACM Symposium on Theory of Computing;2023-06-02
3. Towards an Advanced Deep Learning for the Internet of Behaviors: Application to Connected Vehicles;ACM Transactions on Sensor Networks;2022-12-20
4. Cut Query Algorithms with Star Contraction;2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS);2022-10