Geometric approach to error correcting codes and reconstruction of signals

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.Comment: 17 pages, 3 figure

Error correction via linear programming

Suppose we wish to transmit a vector f Є R^n reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are corrupted in a completely arbitrary fashion. We do not know which entries are affected nor do we know how they are affected. Is it possible to recover f exactly from the corrupted m-dimensional vector y? This paper proves that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ_1 -minimization problem (‖x‖ℓ_1: = ∑_i |xi|) min ‖y − Ag‖ℓ_1 g^∈Rn provided that the fraction of corrupted entries is not too large, i.e. does not exceed some strictly positive constant ρ ∗ (numerical values for ρ ^∗ are given). In other words, f can be recovered exactly by solving a simple convex optimization problem; in fact, a linear program. We report on numerical experiments suggesting that ℓ_1-minimization is amazingly effective; f is recovered exactly even in situations where a very significant fraction of the output is corrupted. In the case when the measurement matrix A is Gaussian, the problem is equivalent to that of counting lowdimensional facets of a convex polytope, and in particular of a random section of the unit cube. In this case we can strengthen the results somewhat by using a geometric functional analysis approach