Low Rank Estimation of Similarities on Graphs

Let (V, E) be a graph with vertex set V and edge set E. Let (X, X', Y) \in V \times V \times {-1, 1} be a random triple, where X, X' are independent uniformly distributed vertices and Y is a label indicating whether X, X' are "similar" (Y = +1), or not (Y = -1). Our goal is to estimate the regression function S\ast (u, v) = E(Y |X = u, X = v), u, v \in V based on training data consisting of n i.i.d. copies of (X, X',Y). We are interested in this problem in the case when S\ast is a symmetric low rank kernel and, in addition to this, it is assumed that S\ast is "smooth" on the graph. We study estimators based on a modified least squares method with complexity penalization involving both the nuclear norm and Sobolev type norms of symmetric kernels on the graph and prove upper bounds on L2 -type errors of such estimators with explicit dependence both on the rank of S\ast and on the degree of its smoothness.