Isometric Embedding and Continuum ISOMAP

Hongyuan Zha, and Zhenyue Zhang

Recently, the Isomap algorithm has been proposed for learning a nonlinear manifold from a set of unorganized high-dimensional data points. It is based on extending the classical multidimensional scaling method for dimension reduction. In this paper, we present a continuous version of Isomap which we call continuum isomap and show that manifold learning in the continuous framework is reduced to an eigenvalue problem of an integral operator. We also show that the continuum isomap can perfectly recover the underlying natural parametrization if the nonlinear manifold can be isometrically embedded onto an Euclidean space. Several numerical examples are given to illustrate the algorithm.

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