On the restricted invertibility problem with an additional orthogonality constraint for random matrices

TL;DR

This paper studies the restricted invertibility problem for random matrices, focusing on selecting large nearly orthogonal column subsets with guaranteed singular value bounds, especially for Gaussian matrices.

Contribution

It introduces a bound on the number of columns that can be selected from Gaussian matrices with an orthogonality constraint to a fixed vector.

Findings

Provides a lower bound for column selection in Gaussian matrices with orthogonality constraints.

Extends restricted invertibility results to include additional orthogonality conditions.

Analyzes worst-case scenarios for the orthogonality constraint in random matrices.

Abstract

The Restricted Invertibility problem is the problem of selecting the largest subset of columns of a given matrix $X$, while keeping the smallest singular value of the extracted submatrix above a certain threshold. In this paper, we address this problem in the simpler case where $X$ is a random matrix but with the additional constraint that the selected columns be almost orthogonal to a given vector $v$. Our main result is a lower bound on the number of columns we can extract from a normalized i.i.d. Gaussian matrix for the worst $v$.