  We study graph-based Laplacian semi-supervised learning at low labeling
rates. Laplacian learning uses harmonic extension on a graph to propagate
labels. At very low label rates, Laplacian learning becomes degenerate and the
solution is roughly constant with spikes at each labeled data point. Previous
work has shown that this degeneracy occurs when the number of labeled data
points is finite while the number of unlabeled data points tends to infinity.
In this work we allow the number of labeled data points to grow to infinity
with the number of labels. Our results show that for a random geometric graph
with length scale $\varepsilon>0$ and labeling rate $\beta>0$, if $\beta
\ll\varepsilon^2$ then the solution becomes degenerate and spikes form, and if
$\beta\gg \varepsilon^2$ then Laplacian learning is well-posed and consistent
with a continuum Laplace equation. Furthermore, in the well-posed setting we
prove quantitative error estimates of $O(\varepsilon\beta^{-1/2})$ for the
difference between the solutions of the discrete problem and continuum PDE, up
to logarithmic factors. We also study $p$-Laplacian regularization and show the
same degeneracy result when $\beta \ll \varepsilon^p$. The proofs of our
well-posedness results use the random walk interpretation of Laplacian learning
and PDE arguments, while the proofs of the ill-posedness results use
$\Gamma$-convergence tools from the calculus of variations. We also present
numerical results on synthetic and real data to illustrate our results.
