Stable signal recovery from incomplete and inaccurate measurements

Suppose we wish to recover a vector x_0 Є R^m (e.g., a digital signal or image) from incomplete and contaminated observations y = Ax_0 + e; A is an n by m matrix with far fewer rows than columns (n « m) and e is an error term. Is it possible to recover x_0 accurately based on the data y? To recover x_0, we consider the solution x^# to the ℓ_(1-)regularization problem min ‖x‖ℓ_1 subject to ‖Ax - y‖ℓ(2) ≤ Є, where Є is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level ‖x^# - x_0‖ℓ_2 ≤ C Є. As a first example, suppose that A is a Gaussian random matrix; then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of x_0; then stable recovery occurs for almost any set of n coefficients provided that the number of nonzeros is of the order of n/[log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.