Nonlinear sampling for sparse recovery

Abstract

Linear sampling of sparse vectors via sensing matrices has been a much investigated problem in the past decade. The nonlinear sampling methods, such as quadratic forms are also studied marginally to include undesired effects in data acquisition devices (e.g., Taylor series expansion up to two terms). In this paper, we introduce customized nonlinear sampling techniques that provide possibility of sparse signal recovery. The main advantage of the nonlinear method over the conventional linear schemes is the reduction in the number of required samples to 2k for recovery of k-sparse signals. We also introduce a low-complexity reconstruction method similar to the annihilating filter in the sampling of signals with finite rate of innovation (FRI). The disadvantage of this nonlinear sampler is its sensitivity to additive noise; thus, it is suitable in scenarios dealing with noiseless data such as network packets, where the data is either noiseless or it is erased. We show that by increasing the number of samples and applying denoising techniques, one can improve the performance. We further introduce a modified version of the proposed method which has strong links with spectral estimation methods and exhibits a more stable performance under noise and numerical errors. Simulation results confirm that this method is much faster than ℓ1-norm minimization routines, widely used in linear compressed sensing and thus much less complex.