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# A Global Geometric Framework for Nonlinear Dimensionality Reduction

Science22 Dec 2000Vol 290, Issue 5500pp. 2319-2323DOI: 10.1126/science.290.5500.2319

## Abstract

Scientists working with large volumes of high-dimensional data, such as global climate patterns, stellar spectra, or human gene distributions, regularly confront the problem of dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. The human brain confronts the same problem in everyday perception, extracting from its high-dimensional sensory inputs—30,000 auditory nerve fibers or 106 optic nerve fibers—a manageably small number of perceptually relevant features. Here we describe an approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set. Unlike classical techniques such as principal component analysis (PCA) and multidimensional scaling (MDS), our approach is capable of discovering the nonlinear degrees of freedom that underlie complex natural observations, such as human handwriting or images of a face under different viewing conditions. In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure.
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Our proof works by showing that for a sufficiently high density (α) of data points, we can always choose a neighborhood size (ε or K) large enough that the graph will (with high probability) have a path not much longer than the true geodesic, but small enough to prevent edges that “short circuit” the true geometry of the manifold. More precisely, given arbitrarily small values of λ1, λ2, and μ, we can guarantee that with probability at least 1 − μ, estimates of the form
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will hold uniformly over all pairs of data points i,j. For ε-Isomap, we require
$No alternative text available$
$No alternative text available$
where r0 is the minimal radius of curvature of the manifold M as embedded in the input space X, s0 is the minimal branch separation of M in X, V is the (d-dimensional) volume of M, and (ignoring boundary effects) ηd is the volume of the unit ball in Euclidean d-space. For K-Isomap, we let ε be as above and fix the ratio (K + 1)/α = ηd(ε/2)d/2. We then require
$No alternative text available$
$No alternative text available$
$No alternative text available$
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In practice, for finite data sets, dG(i,j) may fail to approximate dM(i,j) for a small fraction of points that are disconnected from the giant component of the neighborhood graph G. These outliers are easily detected as having infinite graph distances from the majority of other points and can be deleted from further analysis.
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The Isomap embedding of the hand images is available at Science Online at www.sciencemag.org/cgi/content/full/290/5500/2319/DC1. For additional material and computer code, see .
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In order to evaluate the fits of PCA, MDS, and Isomap on comparable grounds, we use the residual variance 1 – R2(M, DY). DY is the matrix of Euclidean distances in the low-dimensional embedding recovered by each algorithm. M is each algorithm's best estimate of the intrinsic manifold distances: for Isomap, this is the graph distance matrix DG; for PCA and MDS, it is the Euclidean input-space distance matrix DX (except with the handwritten “2”s, where MDS uses the tangent distance). R is the standard linear correlation coefficient, taken over all entries of M and DY.
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In each sequence shown, the three intermediate images are those closest to the points 1/4, 1/2, and 3/4 of the way between the given endpoints. We can also synthesize an explicit mapping from input space X to the low-dimensional embedding Y, or vice versa, using the coordinates of corresponding points {xi, yi} in both spaces provided by Isomap together with standard supervised learning techniques (39).
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Supported by the Mitsubishi Electric Research Laboratories, the Schlumberger Foundation, the NSF (DBS-9021648), and the DARPA Human ID program. We thank Y. LeCun for making available the MNIST database and S. Roweis and L. Saul for sharing related unpublished work. For many helpful discussions, we thank G. Carlsson, H. Farid, W. Freeman, T. Griffiths, R. Lehrer, S. Mahajan, D. Reich, W. Richards, J. M. Tenenbaum, Y. Weiss, and especially M. Bernstein.

## Information & Authors

### Information

#### Published In

Science
Volume 290Issue 550022 December 2000
Pages: 2319 - 2323

#### History

Received: 10 August 2000
Accepted: 21 November 2000

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### Authors

#### Affiliations

Joshua B. Tenenbaum*
Department of Psychology and
Vin de Silva
Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
John C. Langford
Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15217, USA.

#### Notes

*
To whom correspondence should be addressed. E-mail: [email protected]

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##### Published In
Science
Volume 290|Issue 5500
22 December 2000
##### Submission history
Received:10 August 2000
Accepted:21 November 2000
Published in print:22 December 2000
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