Geometric approach to error correcting codes and reconstruction of signals

TL;DR

This paper introduces a geometric functional analysis approach to error correction and signal reconstruction, demonstrating that most linear transformations and measurements enable robust decoding and exact recovery of signals.

Contribution

It establishes that most orthogonal transformations and Gaussian measurements form efficient error correcting codes and allow exact signal reconstruction, improving previous results.

Findings

Most linear orthogonal transformations are effective error correcting codes.

Most Gaussian measurement sets enable exact reconstruction of small-support signals.

Most cube sections form polytopes with maximal lower-dimensional facets.

Abstract

We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes.