On the theoretical analysis of cross validation in compressive sensing

Compressive sensing (CS) is a data acquisition technique that measures sparse or compressible signals at a sampling rate lower than their Nyquist rate. Results show that sparse signals can be reconstructed using greedy algorithms, often requiring prior knowledge such as the signal sparsity or the noise level. As a substitute to prior knowledge, cross validation (CV), a statistical method that examines whether a model overfits its data, has been proposed to determine the stopping condition of greedy algorithms. This paper analyses cross validation in a general compressive sensing framework. Furthermore, we provide both theoretical analysis and numerical simulations for a cross-validation modification of orthogonal matching pursuit, referred to as OMP-CV, which has good performance in sparse recovery.