Recent Progress on Matrix Rigidity -- A Survey

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.