Estimating Low Dimensional Structure in Point Clouds

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This project will develop computationally efficient estimation methods with accompanying theory for the problem of identifying low-dimensional structure in point-cloud data, both low and high dimensional. A canonical example is a noisy sample from a manifold. The investigators will develop minimax lower bounds for the estimation problem and construct estimators that achieve these lower bounds. They will then implement these methods in a practically useful form nd apply them to several important scientific problems.Datasets sometimes contain hidden, low-dimensional structure such as clusters, filaments and low dimensional surfaces. The goal of this project to develop rigorously justified, computationally efficient methods for extracting such structure from data. The developed methods will be applied to a diverse set of problems in astrophysics,seismology, biology, and neuroscience. The project will advance knowledge in several fields including computational geometry, machine learning, and statistics.