  Recently, much of the existing work in manifold learning has been done under
the assumption that the data is sampled from a manifold without boundaries and
singularities or that the functions of interest are evaluated away from such
points. At the same time, it can be argued that singularities and boundaries
are an important aspect of the geometry of realistic data.
  In this paper we consider the behavior of graph Laplacians at points at or
near boundaries and two main types of other singularities: intersections, where
different manifolds come together and sharp "edges", where a manifold sharply
changes direction. We show that the behavior of graph Laplacian near these
singularities is quite different from that in the interior of the manifolds. In
fact, a phenomenon somewhat reminiscent of the Gibbs effect in the analysis of
Fourier series, can be observed in the behavior of graph Laplacian near such
points. Unlike in the interior of the domain, where graph Laplacian converges
to the Laplace-Beltrami operator, near singularities graph Laplacian tends to a
first-order differential operator, which exhibits different scaling behavior as
a function of the kernel width. One important implication is that while points
near the singularities occupy only a small part of the total volume, the
difference in scaling results in a disproportionately large contribution to the
total behavior. Another significant finding is that while the scaling behavior
of the operator is the same near different types of singularities, they are
very distinct at a more refined level of analysis.
  We believe that a comprehensive understanding of these structures in addition
to the standard case of a smooth manifold can take us a long way toward better
methods for analysis of complex non-linear data and can lead to significant
progress in algorithm design.
