Learning functions on unknown manifolds

As more and more complex data sources become available, the analysis of graph and manifold data has become an essential part of various sciences. In this thesis, learning functions through samples on a manifold is investigated. Toward this goal, several problems are studied. First, regularization in Sobolev spaces on manifolds are studied, which can be used to find a smooth function on the data manifold. The regularizer is implemented by the iterated Laplacian, which is a natural tool to study Sobolev spaces on manifolds. This way of regularization generalizes thin plate splines from regular grid on a known domain to random samples on an unknown manifold. Second, we study the asymptotic behavior of graph Laplacian eigenmaps method, which is a finite-dimensional approximation of the infinite-dimensional problem. The analysis shows that the method, as a nonparametric regression method, achieves the optimal integrated mean squares error rate in terms of the intrinsic dimensionality of the manifold. Third, the limit behavior of the graph Laplacian on a manifold boundary and its implications are discussed. Finally, we studied a ranking on data manifold algorithm in information retrieval from a functional analysis point of view, which goes beyond the physics-based intuition.