PDE-Inspired Algorithms for Semi-Supervised Learning on Point Clouds

TL;DR

This paper introduces PDE-inspired algorithms for semi-supervised learning on point clouds, leveraging the connection between discrete and continuum $p$-Dirichlet energies to extend labels effectively.

Contribution

It develops numerical schemes that combine density estimation with PDE methods, establishing consistency in large data limits for kernel and spline density estimations.

Findings

Scheme is consistent with large data limit

Effective label extension via PDE methods

Applicable to kernel and spline density estimations

Abstract

Given a data set and a subset of labels the problem of semi-supervised learning on point clouds is to extend the labels to the entire data set. In this paper we extend the labels by minimising the constrained discrete $p$-Dirichlet energy. Under suitable conditions the discrete problem can be connected, in the large data limit, with the minimiser of a weighted continuum $p$-Dirichlet energy with the same constraints. We take advantage of this connection by designing numerical schemes that first estimate the density of the data and then apply PDE methods, such as pseudo-spectral methods, to solve the corresponding Euler-Lagrange equation. We prove that our scheme is consistent in the large data limit for two methods of density estimation: kernel density estimation and spline kernel density estimation.