Analyzing Cross Validation in Compressed Sensing with Mixed Gaussian And Impulse Measurement Noise With L1 Errors

Abstract

Compressed sensing (CS) involves sampling signals at rates less than their Nyquist rates and attempting to reconstruct them after sample acquisition. Most such algorithms have parameters, for example the regularization parameter in LASSO, which need to be chosen carefully for optimal performance. These parameters can be chosen based on assumptions on the noise level or signal sparsity, but this knowledge may often be unavailable. In such cases, cross validation (CV) can be used to choose these parameters in a purely data-driven fashion. Previous work analyzing the use of CV in CS has been based on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{2}$</tex> cross-validation error with Gaussian measurement noise. But it is well known that the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{2}$</tex> error is not robust to impulse noise and provides a poor estimate of the recovery error, failing to choose the best parameter. Here we propose using the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1}-\mathbf{CV}$</tex> error which provides substantial performance benefits given impulse measurement noise. Most importantly, we provide a detailed theoretical analysis and error bounds for the use of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1}-\mathbf{CV}$</tex> error in CS reconstruction. We show that with high probability, choosing the parameter that yields the minimum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1}-\mathbf{CV}$</tex> error is equivalent to choosing the minimum recovery error (which is not observable in practice). To our best knowledge, this is the first paper which theoretically analyzes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1}$</tex> -based CV in CS.