Departmental Papers (CIS)

Document Type

Conference Paper

Date of this Version

7-4-2004

Comments

Proceedings of the Twenty First International Conference on Machine Learning (ICML 2004), held 4-8 July 2004, Banff, Alberta, Canada.
matlab supplement

Abstract

We investigate how to learn a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. Noting that the kernel matrix implicitly maps the data into a nonlinear feature space, we show how to discover a mapping that unfolds the underlying manifold from which the data was sampled. The kernel matrix is constructed by maximizing the variance in feature space subject to local constraints that preserve the angles and distances between nearest neighbors. The main optimization involves an instance of semidefinite programming---a fundamentally different computation than previous algorithms for manifold learning, such as Isomap and locally linear embedding. The optimized kernels perform better than polynomial and Gaussian kernels for problems in manifold learning, but worse for problems in large margin classification. We explain these results in terms of the geometric properties of different kernels and comment on various interpretations of other manifold learning algorithms as kernel methods.

Keywords

kernels, machine learning

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Date Posted: 27 July 2004

This document has been peer reviewed.