Speaker: Tyrus Berry, George Mason University
Learning manifolds from data
Manifold learning is becoming a key tool in data science. By assuming that data lies on or near an unknown manifold embedded in Euclidean space, we can replace or supplement parametric models by learning the unknown manifold from training data. In particular, algorithms that converge to the Laplace-Beltrami operator of the manifold implicitly represent all the geometric and topological information about the underlying manifold. Moreover, the eigenfunctions of the Laplace-Beltrami operator define a generalized Fourier basis for the space of square integrable functions on the manifold (and allows the definition of more general function spaces). Rigorous estimation of these eigenfunctions yields tools from harmonic analysis for interpolating functions and representing operators on the manifold.
In this talk I introduce a new algorithm which provably converges (in the limit of large data) to a Laplace-Beltrami operator for a large class of non-compact manifolds. This theory generalizes previous results which are all restricted to compact manifolds (uncommon in applications due to noise). These results suggest a natural conformal geometry for a manifold equipped with a sampling measure. This conformal geometry is shown to be obtained in the limit of large data from a new graph construction called Continuous k-Nearest Neighbors (CkNN). Simple examples demonstrate that CkNN captures topological features of a data set better than commonly used graph constructions. We conclude by overviewing the remaining challenges and future directions in manifold learning.
Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, VA 22030-4444
Tel. 703-993-1460, Fax. 703-993-1491