Consistency of Fractional Graph-Laplacian Regularization in Semi-Supervised Learning with Finite Labels

Understand

Laplace learning is a popular machine learning algorithm for finding missing labels from a small number of labelled feature vectors using the geometry of a graph.

• More precisely, Laplace learning is based on minimising a graph-Dirichlet energy, equivalently a discrete Sobolev $\Wkp{2}{1}$ semi-norm, constrained to taking the values of known labels on a given subset.
• The variational problem is asymptotically ill-posed as the number of unlabeled feature vectors goes to infinity for finite given labels due to a lack of regularity in minimisers of the continuum Dirichlet energy in any dimension higher than one.
• In particular, continuum minimisers are not continuous.