Geometric Structure of High-Dimensional Data

Abstract

In applications, a high-dimensional data is given as a discrete set in a Euclidean space. If the points of data are well sampled on a manifold, then the data geometry is inherited from the manifold. Since the underlying manifold is hidden, it is hard to know its geometry by the classical manifold calculus. The data graph is a useful tool to reveal the data geometry. To construct a data graph, we first find the neighborhood system on the data, which is determined by the similarity (or dissimilarity) among the data points. The similarity information of data usually is driven by the application in which the data are used. In this chapter, we introduce the methods for defining the data similarity (or dissimilarity). We also introduce the preliminary spectral graph theory to analyze the data geometry. In Section 1, the construction of neighborhood system on data is discussed. The neighborhood system on a data set defines a data graph, which can be considered as a discrete form of a manifold. In Section 2, we introduce the basic concepts of graphs. In Section 3, the spectral graph analysis is introduced as a tool for analyzing the data geometry. Particularly, the Laplacian on a graph is briefly discussed in this section. Most of the materials in Sections 2 and 3 are found in [1–3].