Testing the manifold hypothesis
Authors:
Charles Fefferman, Sanjoy Mitter and Hariharan Narayanan
Journal:
J. Amer. Math. Soc. 29 (2016), 9831049
MSC (2010):
Primary 62G08, 62H15; Secondary 55R10, 57R40
DOI:
https://doi.org/10.1090/jams/852
Published electronically:
February 9, 2016
MathSciNet review:
3522608
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for testing the existence of a manifold that fits a probability distribution supported in a separable Hilbert space, only using i.i.d. samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution $\mathcal {P}$ supported on the unit ball of a separable Hilbert space $\mathcal {H}$. Let $\mathcal {G}(d, V, \tau )$ be the set of submanifolds of the unit ball of $\mathcal {H}$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $\tau$. Let $\mathcal {L}(\mathcal {M}, \mathcal {P})$ denote the meansquared distance of a random point from the probability distribution $\mathcal {P}$ to $\mathcal {M}$. We obtain an algorithm that tests the manifold hypothesis in the following sense.
The algorithm takes i.i.d. random samples from $\mathcal {P}$ as input and determines which of the following two is true (at least one must be):

There exists $\mathcal {M} \in \mathcal {G}(d, CV, \frac {\tau }{C})$ such that $\mathcal {L}(\mathcal {M}, \mathcal {P}) \leq C {\epsilon }.$

There exists no $\mathcal {M} \in \mathcal {G}(d, V/C, C\tau )$ such that $\mathcal {L}(\mathcal {M}, \mathcal {P}) \leq \frac {\epsilon }{C}.$
The answer is correct with probability at least $1\delta$.
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Additional Information
Charles Fefferman
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
MR Author ID:
65640
Email:
cf@math.princeton.edu
Sanjoy Mitter
Affiliation:
Laboratory for Information and Decision Systems, MIT, Cambridge, Massachusetts 02139
Email:
mitter@mit.edu
Hariharan Narayanan
Affiliation:
Department of Statistics and Department of Mathematics, University of Washington, Seattle, Washington 98195
Email:
harin@uw.edu
Received by editor(s):
March 9, 2014
Received by editor(s) in revised form:
February 3, 2015, and August 9, 2015
Published electronically:
February 9, 2016
Additional Notes:
The first author was supported by NSF grant DMS 1265524, AFOSR grant FA95501210425 and U.S.Israel Binational Science Foundation grant 2014055.
The second author was supported by NSF grant EECS1135843.
Article copyright:
© Copyright 2016
American Mathmatical Society