Sparse signal reconstruction from noisy compressive measurements using cross validation

Compressive sensing is a new data acquisition technique that aims to measure sparse and compressible signals at close to their intrinsic information rate rather than their Nyquist rate. Recent results in compressive sensing show that a sparse or compressible signal can be reconstructed from very few incoherent measurements. Although the sampling and reconstruction process is robust to measurement noise, all current reconstruction methods assume some knowledge of the noise power or the acquired signal to noise ratio. This knowledge is necessary to set algorithmic parameters and stopping conditions. If these parameters are set incorrectly, then the reconstruction algorithms either do not fully reconstruct the acquired signal (underfitting) or try to explain a significant portion of the noise by distorting the reconstructed signal (overfitting). This paper explores this behavior and examines the use of cross validation to determine the stopping conditions for the optimization algorithms. We demonstrate that by designating a small set of measurements as a validation set it is possible to optimize these algorithms and reduce the reconstruction error. Furthermore we explore the trade-off between using the additional measurements for cross validation instead of reconstruction.