@Article{freedman:pami02,
author = {D.\ Freedman},
title = {Efficient simplicial reconstructions of manifolds
from their samples},
journal = {Pattern Analysis and Machine Intelligence, IEEE
Transactions on},
year = 2002,
volume = 24,
number = 10,
pages = {1349--1357},
keywords = {Hilbert spaces, computational complexity,
computational geometry, computer vision, learning
(artificial intelligence), topology,
finite-dimensional differentiable manifold, learned
manifold, manifold learning, sampling density,
simplicial complex, simplicial reconstructions, true
manifold},
abstract = {An algorithm for manifold learning is
presented. Given only samples of a
finite-dimensional differentiable manifold and no a
priori knowledge of the manifold's geometry or
topology except for its dimension, the goal is to
find a description of the manifold. The learned
manifold must approximate the true manifold well,
both geometrically and topologically, when the
sampling density is sufficiently high. The proposed
algorithm constructs a simplicial complex based on
approximations to the tangent bundle of the
manifold. An important property of the algorithm is
that its complexity depends on the dimension of the
manifold, rather than that of the embedding
space. Successful examples are presented in the
cases of learning curves in the plane, curves in
space, and surfaces in space; in addition, a case
when the algorithm fails is analyzed.},
issn = {0162-8828},
annote = {}
}