A Near-Linear-Time Solver for Graph $p$-Laplacian Semi-Supervised Learning via Continuation in $p$

Graph-based semi-supervised learning (SSL) propagates a few labels over a similarity graph by minimizing a Dirichlet-type energy. The standard quadratic ($p=2$) energy reduces to a single graph-Laplacian solve, but it degenerates exactly where SSL is most useful when labels are scarce: gathering more unlabeled data drives the $p=2$ estimate to a near-constant function whenever $d\ge2$ (Nadler-Srebro-Zhou). Well-posedness requires the nonlinear $p$-Laplacian energy with $p>d$. Existing solvers reduce this to a sequence of weighted Laplacian solves, but their reference implementations use a direct sparse factorization or ichol-preconditioned CG instead. Plugging a near-linear Laplacian solver is not straightforward: at large $p$ the conductance weights degenerate near flat-gradient edges, making the system nearly singular and causing stagnation without a damped outer iteration. We close this gap. Recasting $p$-Laplacian SSL as a source-form nonlinear Laplacian flow $Bρ_p(B^\top x)=b$ and solving by damped chord-Newton continuation in $p$, every linearized system stays well-conditioned and can be delegated to a near-linear Laplacian engine. On size-scaled graph families the wall-clock is empirically $m^{0.96}$-$m^{1.02}$ per family (approximate Cholesky default), and a pooled fit across 228 SuiteSparse graphs gives $m^{1.19}$ vs.\ $m^{1.45}$ for direct factorization; the solver handles a $6.8\times10^7$-edge social network in minutes. Memory is the binding constraint: Cholesky fill reaches $10$-$280\times$ the graph nonzeros vs.\ our $O(m)$ hierarchy. Against the released FCL solver we are $1.5$-$14\times$ faster at matched accuracy. On MNIST $10$-NN, $p=3$ scores $64\%$ at one label per class vs.\ $36\%$ for $p=2$. Code: https://github.com/orenlivne/np.