This paper identifies a broad class of nonlinear dimensionality reduced (NLDR) problems where the exact local isometry between an extrinsically curved data manifold M and a low-dimensional parameterization space can be recovered from a finite set of high-dimensional point sampels. The method, Geodesic Nullsapce Analysis (GNA), rests on two results: First, the exact isometric parameterization of a local point clique on M haas an algebraic reduction to arc-length integrations when the ambient-space embedding of M is locally a product of planar quadrics. Second, the locally isometric global parameterization lies in the left invariant subspace of a linearizing operator that averages the nullspace projectors of the local parameterizations. We show how to use the GNA operator for denosing, dimensionality reduction, and resynthesis of both the original data and of new samples, making such s̈ubmanifold̈methods an attractive alternative to subspace methods in data analysis. British Machine Vision Conference (BMVC) This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c ©Mitsubishi Electric Research Laboratories, Inc., 2004 201 Broadway, Cambridge, Massachusetts 02139
[1]
W. T. Tutte.
Convex Representations of Graphs
,
1960
.
[2]
M. Fiedler.
Algebraic connectivity of graphs
,
1973
.
[3]
M. Fiedler.
A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory
,
1975
.
[4]
Johannes Wallner,et al.
On Surface Approximation Using Developable Surfaces
,
1999,
Graph. Model. Image Process..
[5]
J. Tenenbaum,et al.
A global geometric framework for nonlinear dimensionality reduction.
,
2000,
Science.
[6]
Senlin Xu,et al.
SUBMANIFOLDS OF PRODUCT RIEMANNIAN MANIFOLD
,
2000
.
[7]
S T Roweis,et al.
Nonlinear dimensionality reduction by locally linear embedding.
,
2000,
Science.
[8]
Mukund Balasubramanian,et al.
The Isomap Algorithm and Topological Stability
,
2002,
Science.
[9]
Matthew Brand,et al.
Charting a Manifold
,
2002,
NIPS.
[10]
Balázs Kégl,et al.
Intrinsic Dimension Estimation Using Packing Numbers
,
2002,
NIPS.
[11]
Michael J. Kirby,et al.
Estimation of Topological Dimension
,
2003,
SDM.
[12]
Hongyuan Zha,et al.
Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment
,
2002,
ArXiv.
[13]
Nicolas Le Roux,et al.
Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering
,
2003,
NIPS.
[14]
Mikhail Belkin,et al.
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
,
2003,
Neural Computation.
[15]
Kilian Q. Weinberger,et al.
Learning a kernel matrix for nonlinear dimensionality reduction
,
2004,
ICML.
[16]
Bernhard Schölkopf,et al.
A kernel view of the dimensionality reduction of manifolds
,
2004,
ICML.
[17]
Stephen P. Boyd,et al.
The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem
,
2006,
SIAM Rev..