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Recently, much of the existing work in manifold learning has been done underthe assumption that the data is sampled from a manifold without boundaries andsingularities or that the functions of interest are evaluated away from suchpoints. At the same time, it can be argued that singularities and boundariesare an important aspect of the geometry of realistic data.  In this paper we consider the behavior of graph Laplacians at points at ornear boundaries and two main types of other singularities: intersections, wheredifferent manifolds come together and sharp "edges", where a manifold sharplychanges direction. We show that the behavior of graph Laplacian near thesesingularities is quite different from that in the interior of the manifolds. Infact, a phenomenon somewhat reminiscent of the Gibbs effect in the analysis ofFourier series, can be observed in the behavior of graph Laplacian near suchpoints. Unlike in the interior of the domain, where graph Laplacian convergesto the Laplace-Beltrami operator, near singularities graph Laplacian tends to afirst-order differential operator, which exhibits different scaling behavior asa function of the kernel width. One important implication is that while pointsnear the singularities occupy only a small part of the total volume, thedifference in scaling results in a disproportionately large contribution to thetotal behavior. Another significant finding is that while the scaling behaviorof the operator is the same near different types of singularities, they arevery distinct at a more refined level of analysis.  We believe that a comprehensive understanding of these structures in additionto the standard case of a smooth manifold can take us a long way toward bettermethods for analysis of complex non-linear data and can lead to significantprogress in algorithm design.

Graph Laplacians on Singular Manifolds: Toward understanding complex spaces: graph Laplacians on manifolds with singularities and boundaries