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Learning gradients on manifolds. (English) Zbl 1200.62070
Summary: 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.
MSC:
62H99Multivariate analysis
62H30Classification and discrimination; cluster analysis (statistics)
53B99Local differential geometry
65C60Computational problems in statistics