Abstract
We consider algorithms with access to an unknown matrix M in F^{n x d} via matrixvector 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^{i1}. 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 trianglefree. We also show separations for algorithms that are allowed to obtain matrixvector 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 matrixvector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edgevertex incidence matrix) to represent the graph.
Surprisingly, this fundamental model does not appear to have been studied on its own, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.
BibTeX  Entry
@InProceedings{sun_et_al:LIPIcs:2019:10670,
author = {Xiaoming Sun and David P. Woodruff and Guang Yang and Jialin Zhang},
title = {{Querying a Matrix Through MatrixVector Products}},
booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages = {94:194:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771092},
ISSN = {18688969},
year = {2019},
volume = {132},
editor = {Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10670},
URN = {urn:nbn:de:0030drops106709},
doi = {10.4230/LIPIcs.ICALP.2019.94},
annote = {Keywords: Communication complexity, linear algebra, sketching}
}
Keywords: 

Communication complexity, linear algebra, sketching 
Collection: 

46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) 
Issue Date: 

2019 
Date of publication: 

04.07.2019 