## Finding the homology of submanifolds with high confidence from random samples.(English)Zbl 1148.68048

Summary: 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.

### MSC:

 68T05 Learning and adaptive systems in artificial intelligence 60D05 Geometric probability and stochastic geometry 62-07 Data analysis (statistics) (MSC2010)

### Keywords:

data analysis in high-dimensional spaces
Full Text:

### References:

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