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February 2010 Learning gradients on manifolds
Sayan Mukherjee, Qiang Wu, Ding-Xuan Zhou
Bernoulli 16(1): 181-207 (February 2010). DOI: 10.3150/09-BEJ206


A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on manifolds for dimension reduction for high-dimensional data with few observations. We obtain generalization error bounds for the gradient estimates and show that the convergence rate depends on the intrinsic dimension of the manifold and not on the dimension of the ambient space. We illustrate the efficacy of this approach empirically on simulated and real data and compare the method to other dimension reduction procedures.


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Sayan Mukherjee. Qiang Wu. Ding-Xuan Zhou. "Learning gradients on manifolds." Bernoulli 16 (1) 181 - 207, February 2010.


Published: February 2010
First available in Project Euclid: 12 February 2010

zbMATH: 1200.62070
MathSciNet: MR2648754
Digital Object Identifier: 10.3150/09-BEJ206

Keywords: ‎classification‎ , Feature selection , manifold learning , regression , shrinkage estimator , Tikhonov regularization

Rights: Copyright © 2010 Bernoulli Society for Mathematical Statistics and Probability


Vol.16 • No. 1 • February 2010
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