A geometric viewpoint of manifold learning

Abstract

In many data analysis tasks, one is often confronted with very high dimensional data. The manifold assumption, which states that the data is sampled from a submanifold embedded in much higher dimensional Euclidean space, has been widely adopted by many researchers. In the last 15 years, a large number of manifold learning algorithms have been proposed. Many of them rely on the evaluation of the geometrical and topological of the data manifold. In this paper, we present a review of these methods on a novel geometric perspective. We categorize these methods by three main groups: Laplacian-based, Hessian-based, and parallel field-based methods. We show the connection and difference between these three groups on their continuous and discrete counterparts. The discussion is focused on the problem of dimensionality reduction and semi-supervised learning.