Stabilizing Nonuniformly Quantized Compressed Sensing With Scalar Companders

This paper addresses the problem of stably recovering sparse or compressible signals from compressed sensing measurements that have undergone optimal nonuniform scalar quantization, i.e., minimizing the common ℓ 2 -norm distortion. Generally, this quantized compressed sensing (QCS) problem is solved by minimizing the ℓ 1 -norm constrained by the ℓ 2 -norm distortion. In such cases, remeasurement and quantization of the reconstructed signal do not necessarily match the initial observations, showing that the whole QCS model is not consistent. Our approach considers instead that quantization distortion more closely resembles heteroscedastic uniform noise, with variance depending on the observed quantization bin. Generalizing our previous work on uniform quantization, we show that for nonuniform quantizers described by the "compander" formalism, quantization distortion may be better characterized as having bounded weighted ℓ p -norm (p ≥ 2), for a particular weighting. We develop a new reconstruction approach, termed Generalized Basis Pursuit DeNoise (GBPDN), which minimizes the ℓ 1 -norm of the signal to reconstruct constrained by this weighted ℓ p -norm fidelity. We prove that, for standard Gaussian sensing matrices and K sparse or compressible signals in R N with at least Ω((K logN/K) p/2 ) measurements, i.e., under strongly oversampled QCS scenario, GBPDN is ℓ 2 -ℓ 1 instance optimal and stable recovers all such sparse or compressible signals. The reconstruction error decreases as O(2 -B /√(p+1)) given a budget of B bits per measurement. This yields a reduction by a factor √(p+1) of the reconstruction error compared to the one produced by ℓ 2 -norm constrained decoders. We also propose an primal-dual proximal splitting scheme to solve the GBPDN program which is efficient for large-scale problems. Interestingly, extensive simulations testing the GBPDN effectiveness confirm the trend predicted by the theory, that the reconstruction error can indeed be reduced by increasing p, but this is achieved at a much less stringent oversampling regime than the one expected by the theoretical bounds. Besides the QCS scenario, we also show that GBPDN applies straightforwardly to the related case of CS measurements corrupted by heteroscedastic generalized Gaussian noise with provable reconstruction error reduction.