Querying a Matrix through Matrix-Vector Products

doi:10.1145/3470566

	Querying a Matrix through Matrix-Vector Products
	Sun, Xiaoming 1,2; Woodruff, David P.3; Yang, Guang 1,4; Zhang, Jialin 1,2
刊名	ACM TRANSACTIONS ON ALGORITHMS
	2021-10-01
卷号	17 期号:4 页码:19
关键词	Communication complexity linear algebra sketching
ISSN号	1549-6325
DOI	10.1145/3470566
英文摘要	We consider algorithms with access to an unknown matrix M epsilon F-nxd via matrix-vector products, namely, the algorithm chooses vectors v(1),..., v(q), and observes Mv(1),..., Mv(q). Here the vi 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 matrixM; 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 byMis connected or triangle-free. We also show separations for algorithms that are allowed to obtainmatrix-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.
资助项目	National Natural Science Foundation of China[61832003] ; National Natural Science Foundation of China[61872334] ; Strategic Priority Research Program of Chinese Academy of Sciences[XDA27000000] ; K. C. Wong Education Foundation ; National Science Foundation[CCF-181584]
WOS研究方向	Computer Science ; Mathematics
语种	英语
出版者	ASSOC COMPUTING MACHINERY
WOS记录号	WOS:000705407100004
内容类型	期刊论文
源URL	[http://119.78.100.204/handle/2XEOYT63/16945]
专题	中国科学院计算技术研究所
通讯作者	Sun, Xiaoming
作者单位	1.Chinese Acad Sci, Inst Comp Technol, Beijing, Peoples R China 2.Univ Chinese Acad Sci, Beijing, Peoples R China 3.Carnegie Mellon Univ, Pittsburgh, PA 15213 USA 4.Conflux, Beijing, Peoples R China
推荐引用方式 GB/T 7714	Sun, Xiaoming,Woodruff, David P.,Yang, Guang,et al. Querying a Matrix through Matrix-Vector Products[J]. ACM TRANSACTIONS ON ALGORITHMS,2021,17(4):19.
APA	Sun, Xiaoming,Woodruff, David P.,Yang, Guang,&Zhang, Jialin.(2021).Querying a Matrix through Matrix-Vector Products.ACM TRANSACTIONS ON ALGORITHMS,17(4),19.
MLA	Sun, Xiaoming,et al."Querying a Matrix through Matrix-Vector Products".ACM TRANSACTIONS ON ALGORITHMS 17.4(2021):19.