Non-Convex Decoding for Σ∆-Quantized Compressed Sensing

Recently Güntürk et al. showed that Σ∆ quantization is more effective than memoryless scalar quantization (MSQ) when applied to compressed sensing measurements of sparse signals. MSQ with the l decoder recovers an approximation to the original sparse signal with an error proportional to the quantization step size δQ. For an r-th order Σ∆ scheme the reconstruction accuracy can be improved by a factor of (m/k) for any 0 < α < 1 if m & k(logN), with high probability on the measurement matrix. The method requires a preliminary support recovery stage for which r cannot be too large and δQ must be sufficiently small. In this paper, we remove this requirement, showing that the constrained l and l (for sufficiently small τ ) minimization problems subject to a Σ∆-type quantization constraint would approximate the original signal from the Σ∆ quantized measurements with a comparable reconstruction accuracy. We note that these results allow us to achieve root-exponential reconstruction accuracy while using a fixed quantization alphabet.