Graph Laplacian for Semi-Supervised Learning

TL;DR

This paper introduces a novel graph-Laplacian tailored for semi-supervised learning that integrates density and contrastive information, enabling effective spectral clustering with limited labeled data.

Contribution

A new graph-Laplacian designed for SSL that incorporates label information directly, improving the transition from unsupervised to semi-supervised learning.

Findings

Effective semi-supervised spectral clustering demonstrated

Outperforms traditional Laplacian methods in low-label scenarios

Applicable to various SSL problems

Abstract

Semi-supervised learning is highly useful in common scenarios where labeled data is scarce but unlabeled data is abundant. The graph (or nonlocal) Laplacian is a fundamental smoothing operator for solving various learning tasks. For unsupervised clustering, a spectral embedding is often used, based on graph-Laplacian eigenvectors. For semi-supervised problems, the common approach is to solve a constrained optimization problem, regularized by a Dirichlet energy, based on the graph-Laplacian. However, as supervision decreases, Dirichlet optimization becomes suboptimal. We therefore would like to obtain a smooth transition between unsupervised clustering and low-supervised graph-based classification. In this paper, we propose a new type of graph-Laplacian which is adapted for Semi-Supervised Learning (SSL) problems. It is based on both density and contrastive measures and allows the encoding of the labeled data directly in the operator. Thus, we can perform successfully semi-supervised learning using spectral clustering. The benefits of our approach are illustrated for several SSL problems.