Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity

Abstract

A square matrix V is called rigid if every matrix ${V^\prime}$ obtained by altering a small number of entries of V has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this paper, we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field ${\mathbb{F}_2.}$

Stated informally, we conjecture that under a suitable embedding of ${\mathbb{F}_2^n}$ into ${\mathbb{R}^n,}$ vectors arising from a low-dimensional ${\mathbb{F}_2}$-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a low-dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a collection of results establishing weaker forms and special cases of the conjecture above.