Semi-supervised learning with varifold Laplacians

This paper presents varifold learning, a learning framework based on the mathematical concept of varifolds. Different from manifold based methods, our varifold learning framework does not treat data as being sampled from a manifold; but rather, we presume a weaker varifold structure, based upon which we utilize a Grassmannian manifold at each data point, and convert Grassmannian Laplacians to form the varifold Laplacian by applying linear transformations and aggregating over data points. Two algorithms based on the proposed varifold Laplacian, namely varifold Laplacian eigenmaps and varifold transduction are given, together with theoretical convergence results. Experiments are done on toy and real data sets, and our method consistently gives competitive results suggesting its utility.