Overcoming the curse of dimensionality with Laplacian regularization in semi-supervised learning

TL;DR

This paper addresses the challenge of high-dimensional data in semi-supervised learning by analyzing Laplacian regularization and proposing spectral filtering methods that improve scalability and effectiveness.

Contribution

It provides a statistical analysis of Laplacian regularization in high dimensions and introduces spectral filtering techniques with practical kernel-based implementations.

Findings

Spectral filtering methods mitigate the curse of dimensionality.

Kernel methods enable scalable implementation of the proposed techniques.

The analysis reveals conditions for effective semi-supervised learning with Laplacian regularization.

Abstract

As annotations of data can be scarce in large-scale practical problems, leveraging unlabelled examples is one of the most important aspects of machine learning. This is the aim of semi-supervised learning. To benefit from the access to unlabelled data, it is natural to diffuse smoothly knowledge of labelled data to unlabelled one. This induces to the use of Laplacian regularization. Yet, current implementations of Laplacian regularization suffer from several drawbacks, notably the well-known curse of dimensionality. In this paper, we provide a statistical analysis to overcome those issues, and unveil a large body of spectral filtering methods that exhibit desirable behaviors. They are implemented through (reproducing) kernel methods, for which we provide realistic computational guidelines in order to make our method usable with large amounts of data.