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Abstract-Gaussian and quadratic approximations of message passing algorithms on graphs have attracted considerable recent attention due to their computational simplicity, analytic tractability, and wide applicability in optimization and statistical inference problems. This paper presents a systematic framework for incorporating such approximate message passing (AMP) methods in general graphical models. The key concept is a partition of dependencies of a general graphical model into strong and weak edges, with the weak edges representing small, linearizable couplings of variables. AMP approximations based on the Central Limit Theorem can be readily applied to aggregates of many weak edges and integrated with standard message passing updates on the strong edges. The resulting algorithm, which we call hybrid generalized approximate message passing (HyGAMP), can yield significantly simpler implementations of sum-product and max-sum loopy belief propagation. By varying the partition of strong and weak edges, a performance-complexity tradeoff can be achieved. Group sparsity and multinomial logistic regression problems are studied as examples of the proposed methodology.

Abstract-For the problem of multi-class linear classification and feature selection, we propose approximate message passing approaches to sparse multinomial logistic regression (MLR). First, we propose two algorithms based on the Hybrid Generalized Approximate Message Passing (HyGAMP) framework: one finds the maximum a posteriori (MAP) linear classifier and the other finds an approximation of the test-error-rate minimizing linear classifier. Then we design computationally simplified variants of these two algorithms. Next, we detail methods to tune the hyperparameters of their assumed statistical models using Stein's unbiased risk estimate (SURE) and expectationmaximization (EM), respectively. Finally, using both synthetic and real-world datasets, we demonstrate improved error-rate and runtime performance relative to existing state-of-the-art approaches to sparse MLR.

Alcohol (0.5 g/kg bodyweight) was administered in two sessions and placebo in a third session to normal, healthy social drinkers. A control group was administered a mixer (i.e., no alcohol) for each of the three sessions. The heart rate and vagal tone index (V) response patterns were different to alcohol than to the mixer or placebo. The treatments did not differentially influence pulse transit time. The results indicated that the acute effect of a moderate dose of alcohol on the heart is parasympathetically (i.e., vagally) mediated and has no significant direct sympathetic component.