We present a new perspective on graph-based
methods for collaborative ranking for recommender
systems. Unlike user-based or itembased
methods that compute a weighted average
of ratings given by the nearest neighbors, or lowrank
approximation methods using convex optimization
and the nuclear norm, we formulate
matrix completion as a series of semi-supervised
learning problems, and propagate the known ratings
to the missing ones on the user-user or itemitem
graph globally. The semi-supervised learning
problems are expressed as Laplace-Beltrami
equations on a manifold, or namely, harmonic
extension, and can be discretized by a point integral
method. We show that our approach does
not impose a low-rank Euclidean subspace on the
data points, but instead minimizes the dimension
of the underlying manifold. Our method, named
LDM (low dimensional manifold), turns out to be
particularly effective in generating rankings of
items, showing decent computational efficiency
and robust ranking quality compared to state-ofthe-
art methods.