Low Rank Estimation of Similarities on Graphs

TL;DR

This paper proposes a method for estimating a low-rank, smooth similarity function on graphs using a modified least squares approach with nuclear and Sobolev norm penalties, providing theoretical error bounds.

Contribution

It introduces a novel estimator combining nuclear and Sobolev norms for low-rank, smooth kernel estimation on graphs, with proven error bounds.

Findings

Estimator achieves bounds depending on rank and smoothness.

Explicit error bounds are derived for the proposed method.

Method effectively captures low-rank, smooth similarities on graphs.

Abstract

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.