Semi-Supervised Laplacian Learning on Stiefel Manifolds
Chester Holtz Pengwen Chen Alexander Cloninger Chung-Kuan Cheng Gal Mishne
Motivated by the need to address the degeneracy of canonical Laplace learning
algorithms in low label rates, we propose to reformulate graph-based
semi-supervised learning as a nonconvex generalization of a \emph{Trust-Region
Subproblem} (TRS). This reformulation is motivated by the well-posedness of
Laplacian eigenvectors in the limit of infinite unlabeled data. To solve this
problem, we first show that a first-order condition implies the solution of a
manifold alignment problem and that solutions to the classical \emph{Orthogonal
Procrustes} problem can be used to efficiently find good classifiers that are
amenable to further refinement. Next, we address the criticality of selecting
supervised samples at low-label rates. We characterize informative samples with
a novel measure of centrality derived from the principal eigenvectors of a
certain submatrix of the graph Laplacian. We demonstrate that our framework
achieves lower classification error compared to recent state-of-the-art and
classical semi-supervised learning methods at extremely low, medium, and high
label rates. Our code is available on github\footnote{anonymized for
submission}.