Problems of learning on manifolds
This thesis discusses the general problem of learning a function on a manifold given by data points. The space of functions on a Riemannian manifold has a family of smoothness functionals and a canonical basis associated to the Laplace-Beltrami operator. Moreover, the Laplace-Beltrami operator can be reconstructed with certain convergence guarantees when the manifold is only known through the sampled data points. This allows the techniques of regularization and Fourier analysis to be applied to functions defined on data. A convergence result is proved for the case when data is sampled from a compact submanifold of R∧k . Several applications are considered.