From graph to manifold Laplacian: The convergence rate

The convergence of the discrete graph Laplacian to the continuous manifold Laplacian in the limit of sample size N → ∞ while the kernel bandwidth ε → 0, is the justification for the success of Laplacian based algorithms in machine learning, such as dimensionality reduction, semi-supervised learning and spectral clustering. In this paper we improve the convergence rate of the variance term recently obtained by Hein et al. [From graphs to manifolds-Weak and strong pointwise consistency of graph Laplacians, in: P. Auer, R. Meir (Eds.), Proc. 18th Conf. Learning Theory (COLT), Lecture Notes Comput. Sci., vol. 3559, Springer-Verlag, Berlin, 2005, pp. 470-485], improve the bias term error, and find an optimal criteria to determine the parameter ε given N.