Low rank estimation of smooth kernels on graphs

Let (V , A) be a weighted graph with a finite vertex set V , with a symmetric matrix of nonnegative weights A and with Laplacian δ. Let S* :V × V → ℝ be a symmetric kernel defined on the vertex set V . Consider n i.i.d. observations (X_j,X′_j ,Y_j ), j = 1, . . . , n, where X_j,X′_j are independent random vertices sampled from the uniform distribution in V and Y_j ∈ ℝ is a real valued response variable such that (Y _j |X_j,X′_j ) = S*(X _j,X′_j ), j = 1, . . . , n. The goal is to estimate the kernel S* based on the data (X₁,X′₁ ,Y₁), . . . , (X_n,X′_n,Y_n) and under the assumption that S* is low rank and, at the same time, smooth on the graph (the smoothness being characterized by discrete Sobolev norms defined in terms of the graph Laplacian). 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.