Machine Learning (stat.ML)

Abstract: Maximum-Entropy Distributions offer an attractive family of probability densities suitable for moment closure problems. Yet finding the Lagrange multipliers which parametrize these distributions, turns out to be a computational bottleneck for practical closure settings. Motivated by recent success of Gaussian processes, we investigate the suitability of Gaussian priors to approximate the Lagrange multipliers as a map of a given set of moments. Examining various kernel functions, the hyperparameters are optimized by maximizing the log-likelihood. The performance of the devised data-driven Maximum-Entropy closure is studied for couple of test cases including relaxation of non-equilibrium distributions governed by Bhatnagar-Gross-Krook and Boltzmann kinetic equations.

Abstract: Offline change point detection seeks to identify points in a time series where the data generating process changes. This problem is well studied for univariate i.i.d. data, but becomes challenging with increasing dimension and temporal dependence. For the at most one change point problem, we propose the use of a conceptor matrix to learn the characteristic dynamics of a specified training window in a time series. The associated random recurrent neural network acts as a featurizer of the data, and change points are identified from a univariate quantification of the distance between the featurization and the space spanned by a representative conceptor matrix. This model agnostic method can suggest potential locations of interest that warrant further study. We prove that, under mild assumptions, the method provides a consistent estimate of the true change point, and quantile estimates for statistics are produced via a moving block bootstrap of the original data. The method is tested on simulations from several classes of processes, and we evaluate performance with clustering metrics, graphical methods, and observed Type 1 error control. We apply our method to publicly available neural data from rats experiencing bouts of non-REM sleep prior to exploration of a radial maze.