  Given a weighted graph with $N$ vertices, consider a real-valued regression
problem in a semi-supervised setting, where one observes $n$ labeled vertices,
and the task is to label the remaining ones. We present a theoretical study of
$\ell_p$-based Laplacian regularization under a $d$-dimensional geometric
random graph model. We provide a variational characterization of the
performance of this regularized learner as $N$ grows to infinity while $n$
stays constant, the associated optimality conditions lead to a partial
differential equation that must be satisfied by the associated function
estimate $\hat{f}$. From this formulation we derive several predictions on the
limiting behavior the $d$-dimensional function $\hat{f}$, including (a) a phase
transition in its smoothness at the threshold $p = d + 1$, and (b) a tradeoff
between smoothness and sensitivity to the underlying unlabeled data
distribution $P$. Thus, over the range $p \leq d$, the function estimate
$\hat{f}$ is degenerate and "spiky," whereas for $p\geq d+1$, the function
estimate $\hat{f}$ is smooth. We show that the effect of the underlying density
vanishes monotonically with $p$, such that in the limit $p = \infty$,
corresponding to the so-called Absolutely Minimal Lipschitz Extension, the
estimate $\hat{f}$ is independent of the distribution $P$. Under the assumption
of semi-supervised smoothness, ignoring $P$ can lead to poor statistical
performance, in particular, we construct a specific example for $d=1$ to
demonstrate that $p=2$ has lower risk than $p=\infty$ due to the former penalty
adapting to $P$ and the latter ignoring it. We also provide simulations that
verify the accuracy of our predictions for finite sample sizes. Together, these
properties show that $p = d+1$ is an optimal choice, yielding a function
estimate $\hat{f}$ that is both smooth and non-degenerate, while remaining
maximally sensitive to $P$.
