NDLI: A Topological View of Unsupervised Learning from Noisy Data

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Smale, S. Niyogi, P. Weinberger, S.
Copyright Year	2011
Abstract	In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions. A special case of our construction is the mixture of Gaussians where there is Gaussian noise concentrated around a finite set of points (the means). More generally we consider Gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it. We show that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension. Our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial Laplacian of a suitable data-derived simplicial complex.
Starting Page	646
Ending Page	663
Page Count	18
File Format	PDF
ISSN	00975397
DOI	10.1137/090762932
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3 (Special Section on Foundations of Computer Science)
Volume Number	40
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-06-02
Access Restriction	Subscribed
Subject Keyword	Explicit machine computation and programs manifolds Data analysis data Complexity and performance of numerical algorithms topology Computational aspects related to convexity Algebraic topology of manifolds
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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