Nonlinear Blind Compressed Sensing Under Signal-Dependent Noise

In this paper, we consider the problem of nonlinear blind compressed sensing, i.e. jointly estimating the sparse codes and sparsity-promoting basis, under signal-dependent noise. We focus our efforts on the Poisson noise model, though other signal-dependent noise models can be considered. By employing a well-known variance stabilizing transform such as the Anscombe transform, we formulate our task as a nonlinear least squares problem with the ℓ1 penalty imposed for promoting sparsity. We solve this objective function under non-negativity constraints imposed on both the sparse codes and the basis. To this end, we propose a multiplicative update rule, similar to that used in non-negative matrix factorization (NMF), for our alternating minimization algorithm. To the best of our knowledge, this is the first attempt at a formulation for nonlinear blind compressed sensing, with and without the Poisson noise model. Further, we also provide some theoretical bounds on the performance of our algorithm.