A duality view of spectral methods for dimensionality reduction
- Creators
- Xiao, Lin
- Sun, Jun
- Boyd, Stephen
- Others:
- Cohen, William
- Moore, Andrew
Abstract
We present a unified duality view of several recently emerged spectral methods for nonlinear dimensionality reduction, including Isomap, locally linear embedding, Laplacian eigenmaps, and maximum variance unfolding. We discuss the duality theory for the maximum variance unfolding problem, and show that other methods are directly related to either its primal formulation or its dual formulation, or can be interpreted from the optimality conditions. This duality framework reveals close connections between these seemingly quite different algorithms. In particular, it resolves the myth about these methods in using either the top eigenvectors of a dense matrix, or the bottom eigenvectors of a sparse matrix --- these two eigenspaces are exactly aligned at primal-dual optimality.
Additional Information
Copyright 2006 by the author(s)/owner(s). The authors are grateful to Lawrence Saul and Kilian Weinberger for insightful discussions. Part of this work was done when Lin Xiao was on a supported visit at the Institute for Mathematical Sciences, National University of Singapore.Attached Files
Published - p1041-xiao.pdf
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Additional details
- Eprint ID
- 73351
- Resolver ID
- CaltechAUTHORS:20170109-152802886
- Created
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2017-01-10Created from EPrint's datestamp field
- Updated
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2021-11-11Created from EPrint's last_modified field