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Compressed sensing (CS) involves sampling signals at rates less than theirNyquist rates and attempting to reconstruct them after sample acquisition. Mostsuch algorithms have parameters, for example the regularization parameter inLASSO, which need to be chosen carefully for optimal performance. Theseparameters can be chosen based on assumptions on the noise level or signalsparsity, but this knowledge may often be unavailable. In such cases, crossvalidation (CV) can be used to choose these parameters in a purely data-drivenfashion. Previous work analysing the use of CV in CS has been based on the$\ell_2$ cross-validation error with Gaussian measurement noise. But it is wellknown that the $\ell_2$ error is not robust to impulse noise and provides apoor estimate of the recovery error, failing to choose the best parameter. Herewe propose using the $\ell_1$ CV error which provides substantial performancebenefits given impulse measurement noise. Most importantly, we provide adetailed theoretical analysis and error bounds for the use of $\ell_1$ CV errorin CS reconstruction. We show that with high probability, choosing theparameter that yields the minimum $\ell_1$ CV error is equivalent to choosingthe minimum recovery error (which is not observable in practice). To our bestknowledge, this is the first paper which theoretically analyzes $\ell_1$-basedCV in CS.

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