Adaptive Detection of Structured Signals in Low-Rank Interference

In this paper, we consider the problem of detecting the presence (or absence) of an unknown but structured signal from the space-time outputs of an array under strong, non-white interference. Our motivation is the detection of a communication signal in jamming, where often the training portion is known but the data portion is not. We assume that the measurements are corrupted by additive white Gaussian noise of unknown variance and a few strong interferers, whose number, powers, and array responses are unknown. We also assume the desired signals array response is unknown. To address the detection problem, we propose several GLRT-based detection schemes that employ a probabilistic signal model and use the EM algorithm for likelihood maximization. Numerical experiments are presented to assess the performance of the proposed schemes

Binary Linear Classification and Feature Selection via Generalized Approximate Message Passing

For the problem of binary linear classification and feature selection, we propose algorithmic approaches to classifier design based on the generalized approximate message passing (GAMP) algorithm, recently proposed in the context of compressive sensing. We are particularly motivated by problems where the number of features greatly exceeds the number of training examples, but where only a few features suffice for accurate classification. We show that sum-product GAMP can be used to (approximately) minimize the classification error rate and max-sum GAMP can be used to minimize a wide variety of regularized loss functions. Furthermore, we describe an expectation-maximization (EM)-based scheme to learn the associated model parameters online, as an alternative to cross-validation, and we show that GAMP's state-evolution framework can be used to accurately predict the misclassification rate. Finally, we present a detailed numerical study to confirm the accuracy, speed, and flexibility afforded by our GAMP-based approaches to binary linear classification and feature selection

A Simple Derivation of AMP and its State Evolution via First-Order Cancellation

We consider the linear regression problem, where the goal is to recover the vector $\boldsymbol{x}\in\mathbb{R}^n$ from measurements $\boldsymbol{y}=\boldsymbol{A}\boldsymbol{x}+\boldsymbol{w}\in\mathbb{R}^m$ under known matrix $\boldsymbol{A}$ and unknown noise $\boldsymbol{w}$. For large i.i.d. sub-Gaussian $\boldsymbol{A}$, the approximate message passing (AMP) algorithm is precisely analyzable through a state-evolution (SE) formalism, which furthermore shows that AMP is Bayes optimal in certain regimes. The rigorous SE proof, however, is long and complicated. And, although the AMP algorithm can be derived as an approximation of loop belief propagation (LBP), this viewpoint provides little insight into why large i.i.d. $\boldsymbol{A}$ matrices are important for AMP, and why AMP has a state evolution. In this work, we provide a heuristic derivation of AMP and its state evolution, based on the idea of "first-order cancellation," that provides insights missing from the LBP derivation while being much shorter than the rigorous SE proof