Random Matrices and Erasure Robust Frames

TL;DR

This paper proves that random matrices with Gaussian entries are highly robust against data erasures, maintaining stability even when nearly all data vectors are lost, which advances understanding of erasure-resistant data representations.

Contribution

The paper demonstrates that Gaussian random frames are robust against almost complete erasures, establishing the optimal threshold for erasure robustness based on the smallest singular value estimates.

Findings

Random Gaussian frames are robust against nearly 100% erasures.

The robustness threshold depends on the ratio of remaining vectors to the dimension.

A new estimate for the smallest singular value of rectangular Gaussian matrices was developed.

Abstract

Data erasure can often occur in communication. Guarding against erasures involves redundancy in data representation. Mathematically this may be achieved by redundancy through the use of frames. One way to measure the robustness of a frame against erasures is to examine the worst case condition number of the frame with a certain number of vectors erased from the frame. The term {\em numerically erasure-robust frames (NERFs)} was introduced in \cite{FicMix12} to give a more precise characterization of erasure robustness of frames. In the paper the authors established that random frames whose entries are drawn independently from the standard normal distribution can be robust against up to approximately 15\% erasures, and asked whether there exist frames that are robust against erasures of more than 50\%. In this paper we show that with very high probability random frames are, independent of the dimension, robust against any amount of erasures as long as the number of remaining vectors is at least $1+\delta$ times the dimension for some $\delta_0>0$. This is the best possible result, and it also implies that the proportion of erasures can arbitrarily close to 1 while still maintaining robustness. Our result depends crucially on a new estimate for the smallest singular value of a rectangular random matrix with independent standard normal entries.