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 \({\mathbb R}^{d}\).
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Hein, M., Audibert, JY., von Luxburg, U. (2005). From Graphs to Manifolds – Weak and Strong Pointwise Consistency of Graph Laplacians. In: Auer, P., Meir, R. (eds) Learning Theory. COLT 2005. Lecture Notes in Computer Science(), vol 3559. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11503415_32
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DOI: https://doi.org/10.1007/11503415_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26556-6
Online ISBN: 978-3-540-31892-7
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