CDS&E: Geometrical Regression Models Involving Complex Shape Variables

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The need for quantifying shapes of objects arises in many scientific endeavors, with prominent examples in anatomy, biology, physics, and computer vision. These objects can be anatomical parts, biological cells, road networks, facial surfaces, or dinosaur bones. In statistical shape analysis, one uses mathematical representations to measure and analyze statistical variability of shapes within and across subject populations. Furthermore, one studies interactions of shapes with other related variables of interest. For examples, in medical imaging one uses shapes of tumors to diagnose and treat diseases or one studies the effects of aging on shapes of cellular structures to develop appropriate drugs. Such studies are broadly termed shape regression, where one forms statistical models for analyzing interactions of shapes with other variables of interests. Shape can either be used a predictors or responses depending upon the problem context. There is an urgent need to develop formal statistical tools, especially regression models, for analyzing shape data in many disciplines. While recent years have seen tremendous progress in Riemannian approaches to shape representations, the development of statistical models for shape regressions has been relatively limited. The two biggest challenges are non-Euclidean nature of shape representations and lack of registrations in given object data. Past approaches are restricted to statistical models that use pre-registered data and globally linear approximations on one hand, and machine learning solutions that lack interpretable solutions on the other. The proposed research will provide detailed interpretable solutions with ability to formally test relationships between shapes and other variables, even when data is sparse. The key innovations are: (1) use of locally linear approximations of shape manifolds to reduce distortions, and (2) incorporate optimizations over nuisance (registration-related) transformations inside regression models rather than as pre-processing. The project will develop underlying estimation theory and efficient computational solutions for implementing these methods across scientific disciplines. These models will also be amenable to development of real-time, scalable algorithms for analyzing large datasets, for estimation, prediction and testing of models involving shape variables. This project brings together a broad expertise from diverse areas such as computational Riemannian geometry, statistical methodology, and scientific applications, to make new inroads. This research direction, while clearly challenging, represents a fresh perspective and a great opportunity to develop new statistics and to contribute in major scientific advances.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.