The game theoreticp-Laplacian and semi-supervised learning with few labels

Abstract

We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular, we show that the continuum limit of graph-based semi-supervised learning with the game theoretic p-Laplacian is a weighted version of the continuous p-Laplace equation. We also prove that solutions to the graph p-Laplace equation are approximately Hölder continuous with high probability. Our proof uses the viscosity solution machinery and the maximum principle on a graph.

Properly-Weighted Graph Laplacian for Semi-supervised Learning

“…In all other occurrences, γ is defined as in Section 1.1, and in particular we always assume (11) holds.We also use the notation puq x,r "ş Bpx,rq u dx for the average of u over the ball Bpx, rq, and puq r :" puq 0,r . We also assume in this section that Ω has a Lipschitz boundary.…”

“…We now show that trace zero functions can be approximated in H 1 γ pΩq by smooth functions compactly supported away from Γ. Theorem 2.4 (Trace zero functions). Let α ą d´2 and assume γ satisfies (11) and Ω has a Lipschitz boundary. Then u P H 1 γ,0 pΩq if and only if u P H 1 γ pΩq and Trrus " 0.…”

“…The idea behind the p-Laplacian models is that the continuum variational problem is now the p-Dirichlet problem (3) min u ż Ω |∇u| p dx, and for p ą d the Sobolev embedding W 1,p pΩq ãÑ C 0,α pΩq allows the assignment of boundary values at isolated points. The p-Laplacian models, including the p " 8 version, were proven to be well-posed in the limit of infinite unlabeled data and finite labeled data precisely when p ą d in [10,11,41]. The disadvantage of p-Laplacian models is that the nonlinearity renders them more computationally challenging to solve, compared with standard Laplacian regularization.…”

“…Remark 1.2. In practice, one can take (11) to be the definition of the weights on the whole domain Ω. We only need γ to be smooth for a part of our analysis in Section 2.2.…”

“…We only need γ to be smooth for a part of our analysis in Section 2.2. The issue is that since the distance function dpx, Γq is not differentiable (it is only Lipschitz on ΩzΓ if Γ has more than one point), γ cannot be both smooth and globally given by (11). To elaborate, γ appears as part of the diffusion coefficient in the limiting elliptic problem (see Eq.…”