Finding the Homology of Submanifolds with High Confidence from Random Samples

@article{Niyogi2008FindingTH,
  title={Finding the Homology of Submanifolds with High Confidence from Random Samples},
  author={Partha Niyogi and Stephen Smale and Shmuel Weinberger},
  journal={Discrete & Computational Geometry},
  year={2008},
  volume={39},
  pages={419-441}
}
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… CONTINUE READING
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