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Stabilizing Nonuniformly Quantized Compressed Sensing With Scalar Companders

机译：用标量压缩扩展器稳定非均匀量化的压缩传感

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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 $ell _{2}$-norm distortion. Generally, this quantized compressed sensing (QCS) problem is solved by minimizing the $ell _{1}$-norm constrained by the $ell _{2}$-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 $ell _{p}$-norm ($p geqslant 2$), for a particular weighting. We develop a new reconstruction approach, termed Generalized Basis Pursuit DeNoise (GBPDN), which minimizes the $ell _{1}$-norm of the signal to reconstruct constrained by this weighted $ell _{p}$-norm fidelity. We prove that, for standard Gaussian sensing matrices and $K$ sparse or compressible signals in $ BBR ^{N}$ with at least $Omega ((K log N/K)^{p/2})$ measurements, i.e., under strongly oversampled QCS scenario, GBPDN is $ell _{2}-ell _{1}$ instance optimal and stable recovers all such sparse or compressible signals. The reconstruction error decreases as $O(2^{-B}/sqrt {p+1})$ given a budget of $B$ bits per measurement. This yields a reduction by a factor $sqrt {p+1}$ of the reconstruction error compared to the one produced by $ell _{2}$-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.
机译：本文解决了从经过最佳非均匀标量量化（即，使常见的$ ell _ {2} $-范数失真最小化）的压缩传感测量中稳定恢复稀疏或可压缩信号的问题。通常，通过最小化受$ ell _ {2} $范数失真约束的$ ell _ {1} $范数来解决该量化压缩感测（QCS）问题。在这种情况下，重构信号的重新测量和量化不一定与初始观察结果相符，这表明整个QCS模型不一致。相反，我们的方法认为量化失真更类似于异方差均匀噪声，其方差取决于所观察到的量化区间。概括我们先前关于均匀量化的工作，我们表明，对于“压缩扩展器”形式主义描述的非均匀量化器，量化失真可以更好地表征为具有加权加权$ ell _ {p} $-范数（$ p geqslant 2 $），对于特定的权重。我们开发了一种新的重建方法，称为广义基追踪噪点（GBPDN），该方法将信号的$ ell _ {1} $范数最小化，以这种加权的$ ell _ {p} $范数保真度进行约束。我们证明，对于至少具有$ Omega（（K log N / K）^ {p / 2}）$测量的标准高斯传感矩阵和$ K $稀疏或可压缩信号，在$ BBR ^ {N} $中，即，在严重过采样的QCS情况下，GBPDN是$ ell _ {2} -ell _ {1} $实例最佳，并且稳定地恢复了所有此类稀疏或可压缩信号。给定每次测量$ B $位的预算，重建误差随着$ O（2 ^ {-B} / sqrt {p + 1}）$的减小而减小。与由$ ell _ {2} $范数约束的解码器产生的误差相比，这导致重构误差减少了$ sqrt {p + 1} $。我们还提出了一种原始对偶近端拆分方案来解决GBPDN程序，该程序对于大规模问题非常有效。有趣的是，测试GBPDN有效性的大量仿真证实了该理论所预测的趋势，即确实可以通过增加$ p $来减少重构误差，但这是在比理论上所预期的严格得多的过采样机制下实现的。除了QCS方案外，我们还表明GBPDN可以直接应用于CS测量的相关情况，该测量被异方差广义高斯噪声破坏，并且可减少重构误差。

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