Algorithms for $\ell_p$-based semi-supervised learning on graphs

We develop fast algorithms for solving the variational and game-theoretic $p$-Laplace equations on weighted graphs for $p>2$. The graph $p$-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised learning problems with very few labels, where the minimizer of the graph Laplacian becomes degenerate. We present several efficient and scalable algorithms for both the variational and game-theoretic formulations, and present numerical results on synthetic data and on classification and regression problems that illustrate the effectiveness of the $p$-Laplacian for semi-supervised learning with few labels.