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SATA Stochastic Algebraic Topology and Applications

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Technical Report,30 Sep 2011,29 Sep 2016

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The University of Chicago Chicago United States

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This project was devoted mainly to applications of topology primarily to data analysis, but also to some engineering e.g. control problems. Because of noise and uncertain environments, stochasticity is an important element. Topological invariants are robust to some errors in the bulk, but can frequently be highly sensitive to outliers. The work done in this project concerns the amount of data necessary to solve topological inference, even free of noise, and also the nature of errors caused by noise Different kinds of tail behavior have very different implications, and heavy tails are shown to have severe implications for these methods. Also studied is how much data is necessary to compute topological invariants robustly as a complexity theoretical problem and also as an analysis of algorithms problem, and under what kinds of conditions of local featurelessness of the data sometimes called a condition number or feature size The study of critical points is applied to using these methods for inference within machine learning, and the topology of configuration spaces is applied to control problems. Finally, several of the papers studied nontraditional integrals Euler integration which are related to the Gaussian Kinematic Formula, and have earlier been used for target enumeration,

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