Recent Progress on Matrix Rigidity -- A Survey

TL;DR

This survey reviews recent advances in understanding matrix rigidity, highlighting new results on explicit constructions, low rigidity of classical matrices, and connections to other areas in theoretical computer science.

Contribution

It summarizes recent progress in matrix rigidity, including explicit constructions and the understanding of classical matrices' rigidity properties.

Findings

Many classical matrices are now shown to have low rigidity.

Explicit constructions of rigid matrices in complex classes have been developed.

Matrix rigidity has deep connections to communication complexity and data structures.

Abstract

The concept of matrix rigidity was introduced by Valiant(independently by Grigoriev) in the context of computing linear transformations. A matrix is rigid if it is far(in terms of Hamming distance) from any matrix of low rank. Although we know rigid matrices exist, obtaining explicit constructions of rigid matrices have remained a long-standing open question. This decade has seen tremendous progress towards understanding matrix rigidity. In the past, several matrices such as Hadamard matrices and Fourier matrices were conjectured to be rigid. Very recently, many of these matrices were shown to have low rigidity. Further, several explicit constructions of rigid matrices in classes such as $E$ and $P^{NP}$ were obtained recently. Among other things, matrix rigidity has found striking connections to areas as disparate as communication complexity, data structure lower bounds and error-correcting codes. In this survey, we present a selected set of results that highlight recent progress on matrix rigidity and its remarkable connections to other areas in theoretical computer science.