Numerically erasure-robust frames

TL;DR

This paper introduces numerically erasure-robust frames designed for stable signal reconstruction despite noise and adversarial erasures, analyzes various constructions, and discusses the limitations imposed by high erasure rates.

Contribution

The paper proposes the concept of numerically erasure-robust frames and analyzes their properties across different constructions, highlighting fundamental limitations.

Findings

Random frames, equiangular tight frames, and group frames are considered for robustness.

High erasure rates lead to inevitable numerical instability in reconstruction.

Implications for maximal equiangular tight frames and sparse frames are discussed.

Abstract

Given a channel with additive noise and adversarial erasures, the task is to design a frame that allows for stable signal reconstruction from transmitted frame coefficients. To meet these specifications, we introduce numerically erasure-robust frames. We first consider a variety of constructions, including random frames, equiangular tight frames and group frames. Later, we show that arbitrarily large erasure rates necessarily induce numerical instability in signal reconstruction. We conclude with a few observations, including some implications for maximal equiangular tight frames and sparse frames.