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Abstract:
In the machine learning community it is generally believed that
graph Laplacians corresponding to a finite sample of data points
converge to a continuous Laplace operator if the sample size
increases. Even though this assertion serves as a justification for many
Laplacian-based algorithms, so far only some aspects of this claim
have been rigorously proved. In this paper we close this gap by
establishing the strong pointwise consistency of a family of
graph Laplacians with data-dependent weights to some
weighted Laplace operator. Our investigation also
includes the important case where the data lies on a submanifold of
R^d.
