  In manifold learning, algorithms based on graph Laplacians constructed from
data have received considerable attention both in practical applications and
theoretical analysis. In particular, the convergence of graph Laplacians
obtained from sampled data to certain continuous operators has become an active
research topic recently. Most of the existing work has been done under the
assumption that the data is sampled from a manifold without boundary or that
the functions of interests are evaluated at a point away from the boundary.
However, the question of boundary behavior is of considerable practical and
theoretical interest. In this paper we provide an analysis of the behavior of
graph Laplacians at a point near or on the boundary, discuss their convergence
rates and their implications and provide some numerical results. It turns out
that while points near the boundary occupy only a small part of the total
volume of a manifold, the behavior of graph Laplacian there has different
scaling properties from its behavior elsewhere on the manifold, with global
effects on the whole manifold, an observation with potentially important
implications for the general problem of learning on manifolds.
