For naturally occurring data, the dimension of the given input space is often very large while the data themselves have a low intrinsic dimensionality. Spectral kernel methods are non-linear techniques for transforming data into a coordinate system that efficiently reveal the geometric structure in particular, the connectivity of the data. In this talk, we will focus on one particular technique diffusion maps and diffusion coarse-graining; the construction is based on a Markov random walk on the data and offers a general scheme of simultaneously reorganizing and quantizing graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry. We show that clustering in embedding spaces is equivalent to compressing operators and that the quantization distortion in diffusion space bounds the error of compression of the operator, thus giving a rigorous justification and a precise measure of performance of k-means clustering in spectral embedding spaces. We will discuss two particular applications of diffusion coarse-graining: One application is choosing an appropriate set of prototype similar stellar population (SSP) spectra for parameter estimation of star formation history in galaxies. The other example is texture discrimination by a novel geometry-based metric on distributions. (Part of this work is joint with R.R. Coifman, S. Lafon, J. Richards and C. Schafer.)