Inference via low-dimensional couplings

We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a target measure of interest. Direct simulation from the desired measure can then be achieved by pushing forward reference samples through the map. Yet characterizing such a map---e.g., representing and evaluating it---grows challenging in high dimensions. The central contribution of this paper is to establish a link between the Markov properties of the target measure and the existence of low-dimensional couplings, induced by transport maps that are sparse and/or decomposable. Our analysis not only facilitates the construction of transformations in high-dimensional settings, but also suggests new inference methodologies for continuous non-Gaussian graphical models. For instance, in the context of nonlinear state-space models, we describe new variational algorithms for filtering, smoothing, and sequential parameter inference. These algorithms can be understood as the natural generalization---to the non-Gaussian case---of the square-root Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure

Control and estimation of multi-commodity network flow under aggregation

A paradigm put forth by E. Schr\"odinger in 1931/32, known as Schr\"odinger bridges, represents a formalism to pose and solve control and estimation problems seeking a perturbation from an initial control schedule (in the case of control), or from a prior probability law (in the case of estimation), sufficient to reconcile data in the form of marginal distributions and minimal in the sense of relative entropy to the prior. In the same spirit, we consider traffic-flow and apply a Schr\"odinger-type dictum, to perturb minimally with respect to a suitable relative entropy functional a prior schedule/law so as to reconcile the traffic flow with scarce aggregate distributions on families of indistinguishable individuals. Specifically, we consider the problem to regulate/estimate multi-commodity network flow rates based only on empirical distributions of commodities being transported (e.g., types of vehicles through a network, in motion) at two given times. Thus, building on Schr\"odinger's large deviation rationale, we develop a method to identify {\em the most likely flow rates (traffic flow)}, given prior information and aggregate observations. Our method further extends the Schr\"odinger bridge formalism to the multi-commodity setting, allowing commodities to exit or enter the flow field as well (e.g., vehicles to enter and stop and park) at any time. The behavior of entering or exiting the flow field, by commodities or vehicles, is modeled by a Markov chains with killing and creation states. Our method is illustrated with a numerical experiment.Comment: 12 pages, 5 figure