Rates of convergence for Laplacian semi-supervised learning with low labeling rates

We investigate graph-based Laplacian semi-supervised learning at low labeling rates (ratios of labeled to total number of data points) and establish a threshold for the learning to be well posed. Laplacian learning uses harmonic extension on a graph to propagate labels. It is known that when the number of labeled data points is finite while the number of unlabeled data points tends to infinity, the Laplacian learning becomes degenerate and the solutions become roughly constant with a spike at each labeled data point. In this work, we allow the number of labeled data points to grow to infinity as the total number of data points grows. We show that for a random geometric graph with length scale epsilon > 0, if the labeling rate / << epsilon(2), then the solution becomes degenerate and spikes form. On the other hand, if beta >> epsilon(2), then Laplacian learning is well-posed and consistent with a continuum Laplace equation. Furthermore, in the well-posed setting we prove quantitative error estimates of O( epsilon beta-(1/2)) for the difference between the solutions of the discrete problem and continuum PDE, up to logarithmic factors. We also study p-Laplacian regularization and show the same degeneracy result when / << epsilon( p). The proofs of our well-posedness results use the random walk interpretation of Laplacian learning and PDE arguments, while the proofs of the ill-posedness results use gamma -convergence tools from the calculus of variations. We also present numerical results on synthetic and real data to illustrate our results.