Compressed Sensing with Cross Validation

TL;DR

This paper introduces a cross-validation method for compressed sensing that provides sharp error bounds on signal recovery without prior knowledge of the signal's sparsity, using a minimal additional measurement overhead.

Contribution

It proposes a novel cross-validation approach that estimates the recovery error bounds in compressed sensing without knowing the sparsity level beforehand.

Findings

Error bounds can be achieved with high probability using the method.

Numerical bounds on the error between the original and estimated signals are efficiently computed.

The approach has potential applications beyond compressed sensing.

Abstract

Compressed Sensing decoding algorithms can efficiently recover an N dimensional real-valued vector x to within a factor of its best k-term approximation by taking m = 2klog(N/k) measurements y = Phi x. If the sparsity or approximate sparsity level of x were known, then this theoretical guarantee would imply quality assurance of the resulting compressed sensing estimate. However, because the underlying sparsity of the signal x is unknown, the quality of a compressed sensing estimate x* using m measurements is not assured. Nevertheless, we demonstrate that sharp bounds on the error || x - x* ||_2 can be achieved with almost no effort. More precisely, we assume that a maximum number of measurements m is pre-imposed; we reserve 4log(p) of the original m measurements and compute a sequence of possible estimates (x_j)_{j=1}^p to x from the m - 4log(p) remaining measurements; the errors ||x - x*_j ||_2 for j = 1, ..., p can then be bounded with high probability. As a consequence, numerical upper and lower bounds on the error between x and the best k-term approximation to x can be estimated for p values of k with almost no cost. Our observation has applications outside of compressed sensing as well.