Low Rank Estimation of Smooth Kernels on Graphs

We study a problem of estimation of a symmetric low rank kernel defined on a graph under an additional assumption that the kernel is "smooth", the smoothness being defined in terms of Laplacian of the graph. We obtain several results for such problems, including minimax lower bounds on the L2 -error and upper bounds for penalized least squares estimators both with nonconvex and with convex penalties (based on a combination of nuclear norm and Sobolev type norms).