Efficient reconstruction of sparse vectors from quantized observations

Compressive sensing is a recent technique for estimating a sparse vector from a reduced number of observations. Several algorithms have been developed and studied in this context. However, the reduction of number of samples (or the sampling rate) is one aspect of compressive sensing. In fact, the quantization of the observations, which is generally unavoidable in practice (due to the analog-to-digital conversion), could be also understood as another aspect of compressed sensing aiming at reducing the complexity of the sampling device. Moreover, reducing the ADC resolution might be much more beneficial in terms of circuit complexity and power consumption than decreasing the sampling rate. Therefore, we investigate this further aspect of compressive sensing related to the resolution of measurements instead of their number. We first present an efficient message-passing-like iterative algorithm for estimating a vector from quantized linear noisy observations. Contrary to the related work in [1], the algorithm does not require any prior information about the sparse input (such as distribution or norm) and can be applied for arbitrary quantizer resolution. Then, a state evolution analysis is carried out to study the dynamics of the iterative algorithm and can be used to optimize the quantizer characteristic. Finally, some experimental results are provided to demonstrate the validity and performance of the presented algorithm.