Cross Validation in Compressive Sensing and its Application of OMP-CV Algorithm

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 first analyzes cross validation in a general compressive sensing framework and developed general cross validation techniques which could be used to understand CV-based sparse recovery algorithms. Furthermore, we provide theoretical analysis for OMP-CV, a cross validation modification of orthogonal matching pursuit, which has very good sparse recovery performance. Finally, numerical experiments are given to validate our theoretical results and investigate the behaviors of OMP-CV.