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.
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Dimension-free bounds and structural results in communication complexity
The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to perhaps the more common framework in communication complexity where poly-logarithmic dependencies on the number of input bits are tolerated.
Dimension-free bounds are also closely related to structural results, where one seeks to describe the structure of Boolean matrices and functions that have low complexity. We prove such theorems for several communication and query complexity measures as well as various matrix and operator norms. In several other cases we show that such bounds do not exist.
We propose several conjectures, and establish that, in addition to applications in complexity theory, these problems are central to characterization of the idempotents of the algebra of Schur multipliers, and could lead to new extensions of Cohen’s celebrated idempotent theorem regarding the Fourier algebra.
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- Award ID(s):
- 1947546
- NSF-PAR ID:
- 10488558
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Israel Journal of Mathematics
- Volume:
- 253
- Issue:
- 2
- ISSN:
- 0021-2172
- Page Range / eLocation ID:
- 555 to 616
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11856-022-2365-8
