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

by   Oliver M. Crook, et al.

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.



page 19

page 20

page 26


New Nearly-Optimal Coreset for Kernel Density Estimation

Given a point set P⊂ℝ^d, kernel density estimation for Gaussian kernel i...

SCSS-Net: Superpoint Constrained Semi-supervised Segmentation Network for 3D Indoor Scenes

Many existing deep neural networks (DNNs) for 3D point cloud semantic se...

DetMatch: Two Teachers are Better Than One for Joint 2D and 3D Semi-Supervised Object Detection

While numerous 3D detection works leverage the complementary relationshi...

Stationary Density Estimation of Itô Diffusions Using Deep Learning

In this paper, we consider the density estimation problem associated wit...

Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

We investigate identifying the boundary of a domain from sample points i...

Fast multivariate empirical cumulative distribution function with connection to kernel density estimation

This paper revisits the problem of computing empirical cumulative distri...

Spectral Analysis Of Weighted Laplacians Arising In Data Clustering

Graph Laplacians computed from weighted adjacency matrices are widely us...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.