Dequantizing Compressed Sensing: When Oversampling and Non-Gaussian Constraints Combine (0902.2367v4)

Abstract: 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.

• The paper introduces the BPDQₚ decoder, reducing reconstruction error by a factor of √(p+1) in oversampled quantized settings.
• It enforces quantization consistency by ensuring that re-quantized measurements match the original data, overcoming Gaussian noise assumptions.
• Theoretical and numerical analyses validate that oversampling with BPDQₚ improves signal recovery accuracy, promising benefits for imaging and sensor networks.

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

The document presents a comprehensive paper of recovering sparse or compressible signals from uniformly quantized measurements, an extension of traditional compressed sensing (CS) techniques that often assume smooth Gaussian noise rather than quantization distortion. The primary contribution is the introduction of the Basis Pursuit DeQuantizer of moment $p$ (BPDQ$_p$), a new class of convex optimization decoders. These decoders aim to model the quantization distortion more accurately than the well-known Basis Pursuit DeNoise (BPDN) program, which tends to treat the distortion inaccurately as Gaussian noise.

Main Contributions and Findings

1. Quantization Consistency: One novel aspect of the paper is the emphasis on Quantization Consistency (QC), which insists that the requantized measurements of the reconstructed signal match the original quantized measurements. This constraint has been previously suggested but less rigorously implemented.
2. Decoder Performance: Theoretical analysis shows that the BPDQ$_p$ decoder reduces the reconstruction error due to quantization by a factor of $\sqrt{p+1}$, given an oversampled situation. This is because as $p$ grows, the $\ell_p$-norm, representing the fidelity constraint, approaches a form akin to QC.
3. Oversampling Principle: An innovative result here is that BPDQ$_p$ improves its performance in oversampled conditions, which initially might appear counterintuitive given that CS is generally used to minimize measurement counts. However, in practical settings where quantization depth is fixed by hardware limitations, gathering more measurements can actually enhance the reconstruction fidelity, a point the authors emphasize as key to leveraging their approach.
4. Validation on Random Matrices: The paper specifies that Gaussian random matrices satisfy an extended Restricted Isometry Property (RIP) relative to the $\ell_p$-norm, ensuring the stability of signal reconstruction with their approach.
5. Numerical Simulations: The authors report numerical experiments on signal and image reconstructions that validate the theoretical assertions, notably underscoring the reconstruction quality and QC adherence improvements when employing BPDQ$_p$.

Theoretical and Practical Implications

The incorporation of non-Gaussian quantization models in CS frameworks could influence a wide array of practical applications where measurements are inherently quantized due to digital processing restrictions. Furthermore, the findings suggest avenues for improving systems requiring high fidelity in quantized environments, such as MRI imaging or sensor networks, where hardware limitations necessitate accurate decoding from quantized measurements.

Future Research Directions

This work opens several prospective lines for further research. One potential direction includes exploring varying oversampling factors to deduce optimal conditions under which BPDQ$_p$ achieves maximal gains. Additionally, examining more sophisticated sensing matrix constructions to expand the practicality of the RIP$_{p,2}$ criterion across varied signal domains might lead to broader applicability. It would also be worth investigating the adaptation of this approach to non-uniform scalar quantization scenarios, augmenting its utility in real-world processing applications.

In summary, the paper provides a significant advancement in the dequantization of compressed sensing by tackling the challenge with mathematical rigor and demonstrating the viability of their approach through theoretical and experimental validation. The strategic integration of oversampling and non-Gaussian fidelity constraints serves as a stepping stone for high-fidelity signal recovery in constrained quantized environments.