Abstract
Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.
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The main results of this paper were first presented at a conference in honor of John Franks and Clark Robinson at Northwestern University in April 2003. These results were formally written as Technical Report No. TR-2004-08, Department of Computer Science, University of Chicago.
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Niyogi, P., Smale, S. & Weinberger, S. Finding the Homology of Submanifolds with High Confidence from Random Samples. Discrete Comput Geom 39, 419–441 (2008). https://doi.org/10.1007/s00454-008-9053-2
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DOI: https://doi.org/10.1007/s00454-008-9053-2