Abstract : Many modern applications require modeling and analysis of functions on large, high dimensional, unstructured data sets. One may assume that the data lies on a low dimensional manifold, but this manifold is not known. We have extended the diffusion geometry paradigm for these problems to study function approximation on data defined manifolds. Our algorithms are applied successfully to recognition of hand written digits, classification and missing data problems, automatic diagnosis of age related macular disease based on multi--spectral images, and prediction of blood glucose levels. The ideas are applied to other problems, such as analysis of terrain data and solutions of partial differential equations. The scientific barriers include the development of kernel based methods so as to avoid computation of eigenvalues and eigenvectors of large matrices, and quadrature formulas which are guaranteed to work better than the straightforward Monte Carlo integration method.