Diffusion polynomial frames on metric measure spaces

We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L2 space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suitable K-functionals and the frame transforms. The only major condition required is the uniform boundedness of a summability operator. We give sufficient conditions for this to hold in the context of a very general class of metric measure spaces. The theory is illustrated using the approximation of characteristic functions of caps on a dumbell manifold, and applied to the problem of recognition of hand-written digits. Our methods outperforms comparable methods for semi-supervised learning.