New Directions in Statistical Optimal Transport: Theory, Methods and Applications

In many real-world applications, ranging from computer vision and machine learning to causal inference, researchers want to understand how the dynamics of a system would behave if it were placed in a different environment. We address this question by presenting an approach for learning such local counterfactual dynamics across systems described via probability measures using parallel transport in the Wasserstein space. To perform this parallel transport, standard approximations like Schild's ladder are computationally too costly and require strong regularity conditions on the measures, which are not met in most real-world settings. We therefore provide a new approximation scheme for parallel transport in the Wasserstein space based on Jacobi fields. This approach circumvents the shortcomings of Schild's ladder and can be efficiently implemented by solving a continuity equation. In the simple setting of empirical measures on

Euclidean space, it generalizes domain adaptation via optimal transport by also incorporating dynamics. We demonstrate the utility of the approach through applications in deformation transfer in computer vision, dynamic domain adaptation for smooth measures, and external validity in causal inference.

joint work with Ka Yan Chen, Wonjun Choi, and Marcelo Ortiz
 

The empirical Wasserstein distance has been advocated as a natural test statistic for multivariate goodness-of-fit testing, and has been the subject of intensive study in the recent literature on statistical optimal transport. This body of work has characterized the limiting distribution of the empirical Wasserstein distance (and its regularized counterparts) in increasing levels of generality, which enables the construction of asymptotically valid critical values. Despite these methodological advances, theory has lagged behind on characterizing the power of this and related tests against alternatives separated from the null hypothesis in Wasserstein distance, which is typically the intended set of alternatives. This talk will provide some new steps in this direction. We adopt the minimax perspective, and derive the critical radius for goodness-of-fit testing under the Wasserstein distance, subject to various structural assumptions on the set of alternatives. We derive several simple and intuitive tests which are minimax optimal, and we present some surprises regarding the (sub)optimality of various commonly-used test statistics. This talk is based on joint work with Sivaraman Balakrishnan. 

Detecting dynamic patterns of task-specific responses shared across heterogeneous datasets
is an essential and challenging problem in many scientific applications in medical science and
neuroscience. In our motivating example of rodent electrophysiological data, identifying the
dynamical patterns in neuronal activity associated with ongoing cognitive demands and behavior
is key to uncovering the neural mechanisms of memory. One of the greatest challenges
in investigating a cross-subject biological process is that the systematic heterogeneity across
individuals could significantly undermine the power of existing machine learning methods to
identify the underlying biological dynamics. In addition, many technically challenging neurobiological
experiments are conducted on only a handful of subjects where rich longitudinal
data are available for each subject. The low sample sizes of such experiments could further reduce
the power to detect common dynamic patterns among subjects. In this paper, we propose
a novel heterogeneous data integration framework based on optimal transport to extract shared
patterns in complex biological processes. The key advantages of the proposed method are that
it can increase discriminating power in identifying common patterns by reducing heterogeneity
unrelated to the signal by aligning the extracted latent spatiotemporal information across
subjects. Our approach is effective even with a small number of subjects, and does not require
auxiliary matching information for the alignment. In particular, our method can align longitudinal
data across heterogeneous subjects in a common latent space to capture the dynamics of
shared patterns while utilizing temporal dependency within subjects. Our numerical studies on both simulation settings and neuronal activity data indicate that the proposed data integration
approach improves prediction accuracy compared to existing machine learning methods.