Properly-weighted graph Laplacian for semi-supervised learning

TL;DR

This paper introduces a properly-weighted graph Laplacian method for semi-supervised learning that remains well-posed and stable as data size grows, ensuring convergence to a smooth continuum solution.

Contribution

It proposes a novel weighting scheme for graph Laplacian regularization that guarantees stability and convergence in large-sample limits.

Findings

The method remains well-posed with increasing data size.

It converges to the smooth solution of a continuum variational problem.

The approach is fast and easy to implement.

Abstract

The performance of traditional graph Laplacian methods for semi-supervised learning degrades substantially as the ratio of labeled to unlabeled data decreases, due to a degeneracy in the graph Laplacian. Several approaches have been proposed recently to address this, however we show that some of them remain ill-posed in the large-data limit. In this paper, we show a way to correctly set the weights in Laplacian regularization so that the estimator remains well posed and stable in the large-sample limit. We prove that our semi-supervised learning algorithm converges, in the infinite sample size limit, to the smooth solution of a continuum variational problem that attains the labeled values continuously. Our method is fast and easy to implement.