Compressed sensing (CS) demonstrates that sparse signals can be estimated from under-determined linear systems. Distributed CS (DCS) further reduces the number of measurements by considering joint sparsity within signal ensembles. DCS with jointly sparse signals has applications in multi-sensor acoustic sensing, magnetic resonance imaging with multiple coils, remote sensing, and array signal processing. Multi-measurement vector (MMV) problems consider the estimation of jointly sparse signals under the DCS framework. Two related MMV settings are studied. In the first setting, each signal vector is measured by a different independent and identically distributed (i.i.d.) measurement matrix, while in the second setting, all signal vectors are measured by the same i.i.d. matrix. Replica analysis is performed for these two MMV settings, and the minimum mean squared error (MMSE), which turns out to be identical for both settings, is obtained as a function of the noise variance and number of measurements. To showcase the application of MMV models, the MMSE's of complex CS problems with both real and complex measurement matrices are also analyzed. Multiple performance regions for MMV are identified where the MMSE behaves differently as a function of the noise variance and the number of measurements.Belief propagation (BP) is a CS signal estimation framework that often achieves the MMSE asymptotically. A phase transition for BP is identified. This phase transition, verified by numerical results, separates the regions where BP achieves the MMSE and where it is suboptimal. Numerical results also illustrate that more signal vectors in the jointly sparse signal ensemble lead to a better phase transition.

Fast testing can help mitigate the coronavirus disease 2019 pandemic. Despite their accuracy for single sample analysis, infectious diseases diagnostic tools, like RT-PCR, require substantial resources to test large populations. We develop a scalable approach for determining the viral status of pooled patient samples. Our approach converts group testing to a linear inverse problem, where false positives and negatives are interpreted as generated by a noisy communication channel, and a message passing algorithm estimates the illness status of patients. Numerical results reveal that our approach estimates patient illness using fewer pooled measurements than existing noisy group testing algorithms. Our approach can easily be extended to various applications, including where false negatives must be minimized. Finally, in a Utopian world we would have collaborated with RT-PCR experts; it is difficult to form such connections during a pandemic. We welcome new collaborators to reach out and help improve this work!

In this paper, a communication-efficient multi-processor compressed sensing framework based on the approximate message passing algorithm is proposed. We perform lossy compression on the data being communicated between processors, resulting in a reduction in communication costs with a minor degradation in recovery quality. In the proposed framework, a new state evolution formulation takes the quantization error into account, and analytically determines the coding rate required in each iteration. Two approaches for allocating the coding rate, an online back-tracking heuristic and an optimal allocation scheme based on dynamic programming, provide significant reductions in communication costs.Index Terms-lossy compression, multi-processor approximate message passing, rate distortion function.

We study compressed sensing (CS) signal reconstruction problems where an input signal is measured via matrix multiplication under additive white Gaussian noise. Our signals are assumed to be stationary and ergodic, but the input statistics are unknown; the goal is to provide reconstruction algorithms that are universal to the input statistics. We present a novel algorithmic framework that combines: (i) the approximate message passing (AMP) CS reconstruction framework, which solves the matrix channel recovery problem by iterative scalar channel denoising; (ii) a universal denoising scheme based on context quantization, which partitions the stationary ergodic signal denoising into independent and identically distributed (i.i.d.) subsequence denoising; and (iii) a density estimation approach that approximates the probability distribution of an i.i.d. sequence by fitting a Gaussian mixture (GM) model. In addition to the algorithmic framework, we provide three contributions: (i) numerical results showing that state evolution holds for non-separable Bayesian sliding-window denoisers; (ii) an i.i.d. denoiser based on a modified GM learning algorithm;and (iii) a universal denoiser that does not need information about the range where the input takes values from or require the input signal to be bounded. We provide two implementations of our universal CS recovery algorithm with one being faster and the other being more accurate. The two implementations compare favorably with existing universal reconstruction algorithms in terms of both reconstruction quality and runtime. Index Termsapproximate message passing, compressed sensing, Gaussian mixture model, universal denoising.