Abstract:

The study of diffusion operators of manifolds, graphs and "data sets" is useful for the analysis of the structure of the underlying space and of functions on the space. This in turn has many and important applications to disparate fields including partial differential equations, machine learning, dynamical and control systems, data analysis. We discuss recent results that show how to use heat kernels to find good bi- Lipschitz parametrizations of manifolds. We then discuss old and new ideas and algorithms for multiscale analysis associated to such diffusion operators. Given a local operator $T$ on a manifold or a graph, with large powers of low rank, we present a general multiresolution construction for efficiently computing, representing and compressing $T^t$. This allows the computation, to high precision, of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. The dyadic powers of $T$ can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory: we constructscaling functions and wavelet bases associated to this multiresolution analysis, together with the corresponding down sampling operators. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding efficient algorithms. We will sketch motivating applications, which include function approximation, denoising, and learning on data sets, model reduction for complex stochastic dynamical systems, multiscale analysis of Markov chains and Markov decision processes.