Nonlinear Dimension Reduction via Diffusion Maps

Dimension reduction plays a central role in the interpretation and analysis of high-dimensional data. A prominent technique for nonlinear dimension reduction is based on the use of a diffusion map, defined in terms of the transition matrix of an appropriate random walk on a collection of data points. The transition probabilities efficiently capture the structure of the point cloud, allowing for a low-dimensional representation of the original data. In particular, when a point cloud is sampled from a manifold, the diffusion map is expected to retain essential information about the manifold's geometry. We will discuss the effectiveness of diffusion maps in preserving the structure of a point cloud in the setting where the points lie on a closed curve (a one-dimensional manifold). We will consider both the theoretical and computational aspects of this dimension reduction technique.