One-bit compressive sensing with norm estimation

Consider the recovery of an unknown signal ${x}$ from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that ${x}$ is sparse, and that the measurements are of the form $\operatorname{sign}(\langle {a}_i, {x} \rangle) \in \{\pm1\}$. Since such measurements give no information on the norm of ${x}$, recovery methods from such measurements typically assume that $\| {x} \|_2=1$. We show that if one allows more generally for quantized affine measurements of the form $\operatorname{sign}(\langle {a}_i, {x} \rangle + b_i)$, and if the vectors ${a}_i$ are random, an appropriate choice of the affine shifts $b_i$ allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed ${x}$ in the annulus $r \leq \| {x} \|_2 \leq R$, one may estimate the norm $\| {x} \|_2$ up to additive error $\delta$ from $m \gtrsim R^4 r^{-2} \delta^{-2}$ such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors ${x}$ within a Euclidean ball of known radius.Comment: 20 pages, 2 figure

Signal Recovery From 1-Bit Quantized Noisy Samples via Adaptive Thresholding

In this paper, we consider the problem of signal recovery from 1-bit noisy measurements. We present an efficient method to obtain an estimation of the signal of interest when the measurements are corrupted by white or colored noise. To the best of our knowledge, the proposed framework is the pioneer effort in the area of 1-bit sampling and signal recovery in providing a unified framework to deal with the presence of noise with an arbitrary covariance matrix including that of the colored noise. The proposed method is based on a constrained quadratic program (CQP) formulation utilizing an adaptive quantization thresholding approach, that further enables us to accurately recover the signal of interest from its 1-bit noisy measurements. In addition, due to the adaptive nature of the proposed method, it can recover both fixed and time-varying parameters from their quantized 1-bit samples.Comment: This is a pre-print version of the original conference paper that has been accepted at the 2018 IEEE Asilomar Conference on Signals, Systems, and Computer

Quantization and Compressive Sensing

Quantization is an essential step in digitizing signals, and, therefore, an indispensable component of any modern acquisition system. This book chapter explores the interaction of quantization and compressive sensing and examines practical quantization strategies for compressive acquisition systems. Specifically, we first provide a brief overview of quantization and examine fundamental performance bounds applicable to any quantization approach. Next, we consider several forms of scalar quantizers, namely uniform, non-uniform, and 1-bit. We provide performance bounds and fundamental analysis, as well as practical quantizer designs and reconstruction algorithms that account for quantization. Furthermore, we provide an overview of Sigma-Delta ($\Sigma\Delta$) quantization in the compressed sensing context, and also discuss implementation issues, recovery algorithms and performance bounds. As we demonstrate, proper accounting for quantization and careful quantizer design has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing and Its Applications", 201

Hamming Compressed Sensing

Compressed sensing (CS) and 1-bit CS cannot directly recover quantized signals and require time consuming recovery. In this paper, we introduce \textit{Hamming compressed sensing} (HCS) that directly recovers a k-bit quantized signal of dimensional $n$ from its 1-bit measurements via invoking $n$ times of Kullback-Leibler divergence based nearest neighbor search. Compared with CS and 1-bit CS, HCS allows the signal to be dense, takes considerably less (linear) recovery time and requires substantially less measurements ($\mathcal O(\log n)$). Moreover, HCS recovery can accelerate the subsequent 1-bit CS dequantizer. We study a quantized recovery error bound of HCS for general signals and "HCS+dequantizer" recovery error bound for sparse signals. Extensive numerical simulations verify the appealing accuracy, robustness, efficiency and consistency of HCS.Comment: 33 pages, 8 figure

Quantized Compressed Sensing for Partial Random Circulant Matrices

We provide the first analysis of a non-trivial quantization scheme for compressed sensing measurements arising from structured measurements. Specifically, our analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in line with analogous results on Sigma-Delta quantization associated with random Gaussian or subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove that our approach is stable and robust; i.e., the reconstruction error degrades gracefully in the presence of non-quantization noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes as well as a variation of McDiarmid's inequality.Comment: 15 page