Geometry on Probability Spaces

Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However, this avenue has been limited in many areas where calculus is obstructed, as in singular spaces, and in function spaces of functions on a space X where X itself is a function space. Examples of the latter occur in vision and quantum field theory. In vision it would be useful to do analysis on the space of images and an image is a function on a patch. Moreover, in analysis and geometry, the Lebesgue measure and its counterpart on manifolds are central. These measures are unavailable in the vision example and even in learning theory in general. There is one situation where, in the last several decades, the problem has been studied with some success. That is when the underlying space is finite (or even discrete). The introduction of the graph Laplacian has been a major development in algorithm research and is certainly useful for unsupervised learning theory. The approach taken here is to take advantage of both the classical research and the newer graph theoretic ideas to develop geometry on probability spaces. This starts with a space X equipped with a kernel (like a Mercer kernel) which gives a topology and geometry; X is to be equipped as well with a probability measure. The main focus is on a construction of a (normalized) Laplacian, an associated heat equation, diffusion distance, etc. In this setting, the point estimates of calculus are replaced by integral quantities. One thinks of secants rather than tangents. Our main result bounds the error of an empirical approximation to this Laplacian on X.