Asymptotic behavior of $\ell_p$-based Laplacian regularization in semi-supervised learning

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 lp-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 f̂ . From this formulation we derive several predictions on the limiting behavior the d-dimensional function f̂ , 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 ≤ d, the function estimate f̂ is degenerate and “spiky,” whereas for p ≥ d + 1, the function estimate f̂ is smooth. We show that the effect of the underlying density vanishes monotonically with p, such that in the limit p = ∞, corresponding to the so-called Absolutely Minimal Lipschitz Extension, the estimate f̂ 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 = ∞ 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 f̂ that is both smooth and non-degenerate, while remaining maximally sensitive to P .