Geometry and the complexity of matrix multiplication

Abstract

Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.

On the Rank of a Binary Form

“…The case V 1 = · · · = V r = C 2 is of particular interest in quantum computing (see [6]). For the connection between tensor rank and algebraic complexity theory and other problems see [16]. The article [18] relates the rank of forms with the problem of finding polynomial solutions to partial differential equations with constant coefficients.…”

Ideals of varieties parameterized by certain symmetric tensors

“…The problem of tensor decomposition has been studied for many years and by researchers in many scientific areas as Algebraic Geometry (see for example [9,26,27,1,36]), Algebraic Statistic (see [22,19,30]), Phylogenetic [2,4,23,33], Telecommunications [12], Complexity Theory [3,24,25,34], Quantum Computing [6], Psychometrics [8], Chemometrics [5].…”

A note on compressed sensing and the complexity of matrix multiplication

“…The current best algorithm for multiplying two N × N matrices [7] combines tensor product constructions with a result from additive combinatorics due to Salem and D. C. Spencer [17] to derive an algorithm requiring O(N 2.376 ) operations. For a survey of matrix multiplication complexity and related geometry results see [15]. In this paper, we generalize the following theorem of Coppersmith (see [6]).…”