Dequantizing Compressed Sensing: When Oversampling and Non-Gaussian Constraints Combine

In this paper we study the problem of recovering sparse or compressible signals from uniformly quantized measurements. We present a new class of convex optimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment $p$ (BPDQ$_p$), that model the quantization distortion more faithfully than the commonly used Basis Pursuit DeNoise (BPDN) program. Our decoders proceed by minimizing the sparsity of the signal to be reconstructed subject to a data-fidelity constraint expressed in the $\ell_p$-norm of the residual error for $2\leq p\leq \infty$. We show theoretically that, (i) the reconstruction error of these new decoders is bounded if the sensing matrix satisfies an extended Restricted Isometry Property involving the $\ell_p$ norm, and (ii), for Gaussian random matrices and uniformly quantized measurements, BPDQ$_p$ performance exceeds that of BPDN by dividing the reconstruction error due to quantization by $\sqrt{p+1}$. This last effect happens with high probability when the number of measurements exceeds a value growing with $p$, i.e. in an oversampled situation compared to what is commonly required by BPDN = BPDQ$_2$. To demonstrate the theoretical power of BPDQ$_p$, we report numerical simulations on signal and image reconstruction problems.

💡 Research Summary

This paper addresses the fundamental problem of reconstructing sparse or compressible signals from uniformly quantized compressed‑sensing measurements. The authors observe that the widely used Basis Pursuit DeNoise (BPDN) formulation, which enforces an ℓ₂‑norm data‑fidelity constraint, does not faithfully capture the statistical nature of quantization error, which is uniformly distributed and can produce occasional large deviations. To remedy this, they introduce a new family of convex decoders called Basis Pursuit DeQuantizer of moment p (BPDQₚ), defined for any p ∈