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A Global Geometric Framework for Nonlinear Dimensionality Reduction
Joshua B. Tenenbaum,1*Vin de Silva,2John C. Langford3
Scientists working with large volumes of high-dimensional
data, such as global climate patterns, stellar spectra, or
humangene distributions, regularly confront the problem of
dimensionalityreduction: finding meaningful low-dimensional structures
hiddenin their high-dimensional observations. The human brain
confrontsthe same problem in everyday perception, extracting from its
high-dimensionalsensory inputs--30,000 auditory nerve fibers or
106 optic nerve fibers--a manageably small number of
perceptuallyrelevant features. Here we describe an approach to solving
dimensionalityreduction problems that uses easily measured local
metric informationto learn the underlying global geometry of a data
set. Unlikeclassical techniques such as principal component analysis
(PCA)and multidimensional scaling (MDS), our approach is capable ofdiscovering the nonlinear degrees of freedom that underlie complexnatural observations, such as human handwriting or images of aface
under different viewing conditions. In contrast to previousalgorithms
for nonlinear dimensionality reduction, ours efficientlycomputes a
globally optimal solution, and, for an important classof data
manifolds, is guaranteed to converge asymptotically tothe true
structure.
1 Department of Psychology and
2 Department of Mathematics, Stanford University,
Stanford, CA 94305, USA.
3 Department of Computer
Science, Carnegie Mellon University, Pittsburgh, PA 15217, USA.
*
To whom correspondence should be addressed. E-mail:
jbt{at}psych.stanford.edu
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