Asymptotic behavior of ℓp-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 hatf. From this formulation we derive several predictions on the limiting behavior the d-dimensional function hatf, 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 hatf is degenerate and "spiky," whereas for p≥ d+1, the function estimate hatf 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 hatf 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 hatf that is both smooth and non-degenerate, while remaining maximally sensitive to P.