Papers
Spatial Signal Processing via Optimal Transport: Applications to Time Series Analysis and Measures on Graphs
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Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs
The optimal transportation problem defines a geometry of probability measures which leads to a definition for weighted averages (barycenters) of measures, finding application in the machine learning and computer vision communities as a signal processing tool. Here, we implement a barycentric coding model for measures which are supported on a graph, a context in which the classical optimal transport geometry becomes degenerate, by leveraging a Riemannian structure on the simplex induced by a dynamic formulation of the optimal transport problem. We approximate the exponential mapping associated to the Riemannian structure, as well as its inverse, by utilizing past approaches which compute action minimizing curves in order to numerically approximate transport distances for measures supported on discrete spaces. Intrinsic gradient descent is then used to synthesize barycenters, wherein gradients of a variance functional are computed by approximating geodesic curves between the current iterate and the reference measures; iterates are then pushed forward via a discretization of the continuity equation. Analysis of measures with respect to given dictionary of references is performed by solving a quadratic program formed by computing geodesics between target and reference measures. We compare our novel approach to one based on entropic regularization of the static formulation of the optimal transport problem where the graph structure is encoded via graph distance functions, we present numerical experiments validating our approach, and we conclude that intrinsic gradient descent on the probability simplex provides a coherent framework for the synthesis and analysis of measures supported on graphs.
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Wasserstein-based identification of metastable states in time series data via change point detection and segment clustering
Change point detection for time series analysis is a difficult and important problem in applied statistics, for which a variety of approaches have been developed in the past several decades. Here, the Wasserstein metric is employed as a tool for change-point identification in multi-dimensional time series data in order to identify clusters in time series in an unsupervised way. We leverage the simplicity of the optimal transport cost in the 1-dimensional setting to quickly identify both a segmentation (family of change points for a trajectory) and a clustering for the data when the number of segments is much smaller than the number of data points, making no parametric assumptions about the particular distributions involved. Our change point detection method scales linearly in the size of the data and in the dimension of the samples. We test our approach on idealized synthetic data trajectories, as well as real world trajectories coming from the domain of molecular dynamics simulations and underwater acoustics. We find that segmenting these time series via change points obtained by estimating the Wasserstein metric derivative and then clustering the identified segments as measures with similarity measured by the Wasserstein metric, successfully identifies metastable states in the law of the processes.
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Regularized 2-Wasserstein Barycenters on Discrete Metric Spaces
We apply entropically regularized Wasserstein geometry to the setting of a discrete metric space, where the cost functions are given by different choices of graph metric, which capture varying levels of connectivity information. Despite the degeneracy of the Monge-Kantorovich problem in the case of discrete metric spaces, regularized transport leads to geometric objects (e.g., geodesics and more generally barycenters) which convincingly resemble what one would expect from a Wasserstein-type geometry on the space of probability distributions defined for a network. We consider both synthesis (combining measures with respect to the regularized Wasserstein geometry) and analysis (decomposing signals with respect to fixed reference measures) on real geography data, demonstrating the utility of our approach.
Slides for the talks that go with these are under Talks.