Interlacing Families III: Sharper Restricted Invertibility Estimates

TL;DR

This paper employs interlacing families of polynomials to provide a simpler proof of the Restricted Invertibility Principle and improves the bounds by using Schatten 4-norm stable rank and analyzing polynomial zeros.

Contribution

It introduces sharper bounds for the Restricted Invertibility Principle using advanced polynomial analysis and stable rank concepts.

Findings

Replaces stable rank with Schatten 4-norm stable rank for tighter bounds

Provides a simpler proof of the Restricted Invertibility Principle

Derives bounds from analysis of zeros of Jacobi and Laguerre polynomials

Abstract

We use the method of interlacing families of polynomials to derive a simple proof of Bourgain and Tzafriri's Restricted Invertibility Principle, and then to sharpen the result in two ways. We show that the stable rank can be replaced by the Schatten 4-norm stable rank and that tighter bounds hold when the number of columns in the matrix under consideration does not greatly exceed its number of rows. Our bounds are derived from an analysis of smallest zeros of Jacobi and associated Laguerre polynomials.