Study on the Low-Dimensional Embedding and the Embedding Dimensionality of Manifold of High-Dimensional Data

Abstract

Finding meaningful low-dimensional embedded in a high-dimensional space is a classical problem. Isomap is a nonlinear dimensionality reduction method proposed and based on the theory of manifold. It not only can reveal the meaningful low-dimensional structure hidden in the high-dimensional observation data, but can recover the underlying parameter of data lying on a low-dimensional submanifold. Based on the hypothesis that there is an isometric mapping between the data space and the parameter space, Isomap works, but this hypothesis has not been proved. In this paper, the existence of isometric mapping between the manifold in the high-dimensional data space and the parameter space is proved. By distinguishing the intrinsic dimensionality of high-dimensional data space from the manifold dimensionality, and it is proved that the intrinsic dimensionality is the upper bound of the manifold dimensionality in the high-dimensional space in which there is a toroidal manifold. Finally an algorithm is proposed to find the underlying toroidal manifold and judge whether there exists one. The results of experiments on the multi-pose three-dimensional object show that the method is effective.