Highly robust error correction by convex programming

TL;DR

This paper introduces convex programming-based decoding schemes that enable highly robust error correction in communication systems, effectively recovering transmitted signals despite gross and small errors, with practical algorithms and strong simulation results.

Contribution

It presents two novel convex optimization decoding methods that achieve near-perfect recovery under severe gross errors, advancing error correction techniques with practical convex programs.

Findings

Decoding schemes recover signals with high accuracy despite gross errors.

Convex programming methods are simple and computationally efficient.

Numerical simulations demonstrate excellent practical performance.

Abstract

This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x in R^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrupted by arbitrary gross errors, and when in addition, all the entries of the codeword are contaminated by smaller errors (e.g. quantization errors). We show that if one encodes the information as Ax where A is a suitable m by n coding matrix (m >= n), there are two decoding schemes that allow the recovery of the block of n pieces of information x with nearly the same accuracy as if no gross errors occur upon transmission (or equivalently as if one has an oracle supplying perfect information about the sites and amplitudes of the gross errors). Moreover, both decoding strategies are very concrete and only involve solving simple convex optimization programs, either a linear program or a second-order cone program. We complement our study with numerical simulations showing that the encoder/decoder pair performs remarkably well.