  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.
