Constrained Statistical Estimation and Inference: Theory, Algorithms and Applications

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This project lies at the boundary of statistics and machine learning. The underlying theme is to exploit constraints that are present in complex scientific data analysis problems, but that have not been thoroughly studied in traditional approaches. The project will explore theory, algorithms, and applications of statistical procedures, with constraints imposed on the storage, runtime, shape, energy or physics of the estimators and applications. The overall goal of the research is to develop theory and tools that can help scientists to conduct more effective data analysis.Many statistical methods are purely "data driven" and only place smoothness or regularity restrictions on the underlying model. In particular, classical statistical theory studies estimators without regard to their computational requirements. In modern data analysis settings, including astronomy, cloud computing, and embedded devices, computational demands are often central. The project will develop minimax theory and algorithms for nonparametric estimation and detection problems under constraints on storage, computation, and energy. Other constraints to be studied include shape restrictions such as convexity and monotonicity for high dimensional data. The project will also investigate the incorporation of physical constraints through the use of PDEs and models of physical dynamics and mechanics, focusing on both algorithms and theoretical bounds.