Stochastic Particle Flow for Nonlinear High-Dimensional Filtering Problems

A series of novel filters for probabilistic inference that propose an alternative way of performing Bayesian updates, called particle flow filters, have been attracting recent interest. These filters provide approximate solutions to nonlinear filtering problems. They do so by defining a continuum of densities between the prior probability density and the posterior, i.e. the filtering density. Building on these methods' successes, we propose a novel filter. The new filter aims to address the shortcomings of sequential Monte Carlo methods when applied to important nonlinear high-dimensional filtering problems. The novel filter uses equally weighted samples, each of which is associated with a local solution of the Fokker-Planck equation. This hybrid of Monte Carlo and local parametric approximation gives rise to a global approximation of the filtering density of interest. We show that, when compared with state-of-the-art methods, the Gaussian-mixture implementation of the new filtering technique, which we call Stochastic Particle Flow, has utility in the context of benchmark nonlinear high-dimensional filtering problems. In addition, we extend the original particle flow filters for tackling multi-target multi-sensor tracking problems to enable a comparison with the new filter

Asymptotically Normal Estimation of Local Latent Network Curvature

Network data, commonly used throughout the physical, social, and biological sciences, consist of nodes (individuals) and the edges (interactions) between them. One way to represent the complex, high-dimensional structure in network data is to embed the graph into a low-dimensional geometric space. Curvature of this space, in particular, provides insights about structure in the graph, such as the propensity to form triangles or present tree-like structure. We derive an estimating function for curvature based on triangle side lengths and the midpoints between sides where the only input is a distance matrix and also establish asymptotic normality. We next introduce a novel latent distance matrix estimator for networks as well as an efficient algorithm to compute the estimate via solving iterative quadratic programs. We apply this method to the Los Alamos National Laboratory Unified Network and Host dataset and show how curvature estimates can be used to detect a red-team attack faster than naive methods, as well as discover non-constant latent curvature in coauthorship networks in physics.Comment: 77 page