Finding the Homology of Submanifolds with High Confidence from Random Samples
Partha Niyogi; Stephen Smale; Shmuel Weinberger. 12 November, 2004.
Communicated by Partha Niyogi.
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 is 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
question where the data is noisy and lies near rather than on the submanifold in question.
The original document is available in Postscript (uploaded 12 November, 2004 by