Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Behavior of Graph Laplacians on Manifolds with Boundary   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.

12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph