Performance Limits With Additive Error Metrics in Noisy Multimeasurement Vector Problems

Abstract

Real-world applications such as magnetic resonance imaging with multiple coils, multi-user communication, and diffuse optical tomography often assume a linear model where several sparse signals sharing common sparse supports are acquired by several measurement matrices and then contaminated by noise. Multi-measurement vector (MMV) problems consider the estimation or reconstruction of such signals. In different applications, the estimation error that we want to minimize could be the mean squared error or other metrics such as the mean absolute error and the support set error. Seeing that minimizing different error metrics is useful in MMV problems, we study information-theoretic performance limits for MMV signal estimation with arbitrary additive error metrics. We also propose a message passing algorithmic framework that achieves the optimal performance, and rigorously prove the optimality of our algorithm for a special case. We further conjecture the optimality of our algorithm for some general cases, and back it up through numerical examples. As an application of our MMV algorithm, we propose a novel setup for active user detection in multi-user communication and demonstrate the promise of our proposed setup.

Performance Analysis of Joint Active User Detection and Channel Estimation for Massive Connectivity

“…We call the function (7) as free entropy function 4 . According to [12], [13], optimizing E that maximizes the free entropy function (7) corresponds to the minimum MSE (MMSE) of specific choices of system parameters, i.e., ρ, α, C g , ∀g, in the Bayes-optimal condition. Setting the first order derivative of Φ(E) with respect to the matrix E into zero, we get the following equation.…”

“…In the statistical physics literature, the free entropy of a system reflects the macro performance that characterizes some thermodynamic properties of the system [15]. Some related works in the signal processing literature have also shown that evaluating the fixed point of free entropy function provides the minimum MSE (MMSE) prediction for signal recovery and the achievable MSE for the AMP framework [12], [13]. For the massive connectivity scenario, evaluating the free entropy function of our system (1) provides an analytic tool to measure the performances metrics of joint AUD and CE.…”

“…The region of the system parameters where the AMP framework is sub-optimal cannot be measured via the state evolution analysis in [6], [10]. Finally, some related works [12], [13] have shown that the performance of AMP exhibits a phase transition phenomenon where the AMP algorithm will exhibit disconnect performance variations with variations of the system parameters, which has not been analyzed for massive connectivity.…”

Noisy Pooled PCR for Virus Testing

“…In words, the ROC captures trade-offs between false positives and negatives, and increasing the AUC reflects better estimation. While standard GAMP minimizes the MSE [7], other error metrics can be minimized [10], [11].…”

Noisy Pooled PCR for Virus Testing

“…In words, the ROC captures trade-offs between false positives and negatives, and increasing the AUC reflects better estimation. While standard GAMP minimizes the MSE [7], other error metrics can be minimized [10], [11]. The top pannel of Fig.…”