Functional Regression and Classification for Data Supported on Complex Geometries

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This project aims to develop novel statistical methods for functional/imaging data that are located on complex geometries. Thanks to advances in imaging technology, such data are now ubiquitous in the fields of medicine, biology, and climatology, among others. Perhaps the most common task is to use these complex data (for example, brain activity on the cortical surface) to predict a response variable (for example, age or disease status) by employing regression and classification models. The project will develop a novel framework for regression and classification that leverages mathematical tools including partial differential equations and differential geometry to integrate additional information available and define more accurate models. Moreover, efficient computational implementations and theoretical guarantees on the models' performance on previously unseen data will also be provided. The software implementations of the new models will be made publicly available. The research activities will offer numerous opportunities for interdisciplinary research training of the next generation of statisticians and data scientists.The framework under development will generalize current functional data methodology to complex settings that arise from the analysis of modern imaging data. Specifically, the research activities include development of regularized linear models and generalized linear models for predictors that are functional data supported on multidimensional non-linear domains. These models will be formulated as an infinite-dimensional minimization over spaces of smooth functions. To efficiently approximate their solutions, the project will employ tools from numerical analysis of partial differential equations and elements of calculus of variations. Further, the new framework will be generalized to situations where the functional predictors display tensor structure, which can be leveraged to control the complexity of the solution via low-rank constraints.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.