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

Vivien Cabannes, Loucas Pillaud-Vivien, Francis Bach, Alessandro Rudi

Keywords: semi-supervised learning, Laplacian, Laplacian regularization, weakly supervised learning, kernel methods, spectral filtering, partial labelling, diffusion, functional analysis, statistical learning, generalization bounds

TL;DR: Laplacian regularization the right way

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 design a new class of algorithms overcoming this issue, unveiling a large body of spectral filtering methods. Additionally, we provide a statistical analysis showing that our estimators 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.

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Code: https://github.com/VivienCabannes/partial_labelling