Geometric approach to error-correcting codes and reconstruction of signals

We develop an approach through geometric functional analysis to reconstruction of signals from few linear measurements and to error-correcting codes. 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 show that most linear orthogonal transformations Q: ℝn → ℝ^m form efficient and 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 ℓ1-norm. This yields robust error-correcting codes over reals (and over alphabets of polynomial size), with a Gilbert-Varshamov type bound, and with quadratic time encoders and polynomial time decoders. 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 L1-norm minimization. This is an improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of “neighborly” symmetric polytopes, that is, polytopes with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of a cube form such polytopes. Our work thus belongs to a common ground of coding theory, signal processing, combinatorial geometry, and geometric functional analysis. Our argument, which is based on concentration of measure and improving Lipschitzness by random projections, may be of independent interest in geometric functional analysis.