Work

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

Year: 2019

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.1909.10221

Authors: Crook, Oliver M., Hurst, Tim, Schönlieb, Carola-Bibiane, Thorpe, Matthew, Zygalakis, Konstantinos C.

Institutions: University of Cambridge, MRC Biostatistics Unit, Maxwell Institute for Mathematical Sciences, University of Edinburgh

Language: English

Topic: Machine Learning and Algorithms

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: Heriot-Watt University, Wellcome Trust, Alan Turing Institute, Leverhulme Trust, Scottish Funding Council, European Commission, Goddard Space Flight Center, Engineering and Physical Sciences Research Council

Awards: G1309768829, G1633672591, G1755984640, G2164220759, G2500306150, Nonlocal Methods for Arbitrary Data Sources, G3855435662, G444630160, The Alan Turing Institute, CHallenges in Preservation of Structure, G5639283639, Efficient computational tools for inverse imaging problems, EPSRC Centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging, G7393500401, G8597009950

OpenAlex ID: W2973571117

Cites: 75

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