Abstract

The diffusion map is a dimensional reduction algorithm used for finding the low-dimensional structure of high-dimensional data based on concepts from Riemannian geometry. The diffusion map relies on a geometric assumption which assumes that the data set lies near a low-dimensional smooth manifold embedded in a high-dimensional Euclidean space. The work of Coifman and Lafon [1] shows that using only the data without other prior information about the unknown manifold structure, the data-driven diffusion kernel converges to the Laplace-Beltrami operator on the manifold in the limit of large data. Since the Laplace-Beltrami operator determines the Riemannian metric, the diffusion map can be applied to capture important aspects of the manifold from the data. In this talk, I will give a comprehensive overview of diffusion maps including the key theoretical results and some applications as time permits.

Reference

R. R. Coifman and S. Lafon. (2006). Diffusion maps. Applied and computational harmonic analysis, 21(1), 5-30.