An introduction to diffusion maps

Abstract

This paper describes a mathematical technique [1] for dealing with dimensionality reduction. Given data in a high-dimensional space, we show how to find parameters that describe the lower-dimensional structures of which it is comprised. Unlike other popular methods such as Principle Component Analysis and Multi-dimensional Scaling, diffusion maps are non-linear and focus on discovering the underlying manifold (lower-dimensional constrained “surface” upon which the data is embedded). By integrating local similarities at different scales, a global description of the data-set is obtained. In comparisons, it is shown that the technique is robust to noise perturbation and is computationally inexpensive. Illustrative examples and an open implementation are given.