CAREER: Geometric approaches to hierarchical and nonparametric model-based inference

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Hierarchical and nonparametric models present some of the most fundamental and powerful tools in modern statistics. Despite valuable advances made in the past decades, there are several widely recognized and emerging problems. First, even as these models are increasingly applied to large data sets and complex domains, a statistical theory for inferential behaviors of the hierarchy of latent variables present in the models is not yet available. Second, local inference methods based on sampling, although simple to derive, tend to converge too slowly, thereby losing their effectiveness. Third, most existing methods are incapable of handling highly distributed data sources, which are increasingly responsible for the influx of big data. Addressing these challenges requires fundamentally new ideas in theory, modeling and algorithms that must account for the contrast and interplay between the global geometry of an inference problem and the need for decentralization of inference and algorithmic implementation. This project aims to make fundamental contributions toward advancing hierarchical model-based inference. They include a statistical theory for the latent hierarchy of variables and for analyzing the effects of transfer learning. They also include variational inference algorithms based on the global geometry of latent structures and geometric analyses of the tradeoffs between statistical and computational efficiencies. Both the algorithms and theory are unified by the use of Wasserstein geometry, which arises from the mathematical theory of optimal transportation. Moreover, scalable hierarchical models will be developed that can exploit highly distributed data sources and decentralized inference architectures.This research will improve our ability to manage, analyze and make decisions with large-scale, high dimensional and complex data, especially in the research and applications of networks and the environment. The decentralized detection algorithms for highly distributed data sources have the potential of advancing the state of the art technologies that support data-driven and high-performance distributed computing architectures. As such, this research has the potential of extending the capabilities of the real-time detection and tracking devices currently deployed in the health-care and security domains. The optimal transport based theory will deepen our understanding of hierarchical Bayesian inference, a fundamental concept of modern statistics. The algorithms and geometric analyses will provide useful tradeoffs between statistical and computational complexity, an important issue lying in the interface of Statistics and Computer Science. This research will also provide support for broadening the current statistics curriculum at the University of Michigan. The PI will integrate the teaching of statistical and computational tools with modern applications, by developing synthesis courses which interact closely with research topics of the project. This provides an excellent opportunity to train students with a broad base of knowledge and cross-disciplinary skills in the fields of statistics, probability, machine learning, distributed computation and networked systems.