Discrete & Computational Geometry

, Volume 39, Issue 1, pp 419–441

Finding the Homology of Submanifolds with High Confidence from Random Samples


    • Departments of Computer Science and StatisticsUniversity of Chicago
  • Stephen Smale
    • Toyota Technological Institute
  • Shmuel Weinberger
    • Department of MathematicsUniversity of Chicago

DOI: 10.1007/s00454-008-9053-2

Cite this article as:
Niyogi, P., Smale, S. & Weinberger, S. Discrete Comput Geom (2008) 39: 419. doi:10.1007/s00454-008-9053-2


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.

Copyright information

© Springer Science+Business Media, LLC 2008