Robustness Properties of Dimensionality Reduction with Gaussian Random Matrices

TL;DR

This paper investigates the robustness of Gaussian random matrices in dimensionality reduction, focusing on their ability to preserve norms despite row erasures, and establishes optimal bounds and properties relevant to compressed sensing.

Contribution

It provides the first optimal estimates of erasure ratios for norm preservation and extends the Johnson-Lindenstrauss and restricted isometry properties to corrupted Gaussian matrices.

Findings

Optimal erasure ratio estimates for norm preservation.

Robust Johnson-Lindenstrauss lemma under erasures.

Strong restricted isometry property with near-optimal constants.

Abstract

In this paper we study the robustness properties of dimensionality reduction with Gaussian random matrices having arbitrarily erased rows. We first study the robustness property against erasure for the almost norm preservation property of Gaussian random matrices by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate. As a consequence, we establish the robustness property of Johnson-Lindenstrauss lemma and the robustness property of restricted isometry property with corruption for Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal lower and upper bounds of norm distortion rates of Gaussian random matrices under a given erasure ratio. This allows us to establish the strong restricted isometry property with the almost optimal RIP constants, which plays a central role in the study of phaseless compressed sensing.