An empirical eigenvalue-threshold test for sparsity level estimation from compressed measurements

Compressed sensing allows for a significant reduction of the number of measurements when the signal of interest is of a sparse nature. Most computationally efficient algorithms for signal recovery rely on some knowledge of the sparsity level, i.e., the number of non-zero elements. However, the sparsity level is often not known a priori and can even vary with time. In this contribution we show that it is possible to estimate the sparsity level directly in the compressed domain, provided that multiple independent observations are available. In fact, one can use classical model order selection algorithms for this purpose. Nevertheless, due to the influence of the measurement process they may not perform satisfactorily in the compressed sensing setup. To overcome this drawback, we propose an approach which exploits the empirical distributions of the noise eigenvalues. We demonstrate its superior performance compared to state-of-the-art model order estimation algorithms numerically.