*Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.*

Thursday, March 15 , 2018

5:10 p.m.

BA1180

PhD Candidate: Nan Wu

Co-Supervisors: Alex Nabutovsky, Hau-tieng Wu

Thesis title: Differential Geometry Approach for Unsupervised Machine Learning Algorithms

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Abstract:

Since its introduction in 2000, locally linear embedding (LLE) algorithm has been widely applied in data science.

In this thesis, we provide an asymptotical analysis of LLE under the manifold setup. First, by study the regularized barycentric problem, we derive the corresponding kernel function of LLE. Second, we show that when the point cloud is sampled from a general closed manifold, asymptotically LLE algorithm does not always recover the Laplace-Beltrami operator, and the result may depend on the non-uniform sampling. We demonstrate that a careful choosing of the regularization is necessary to ensure the recovery of the Laplace-Beltrami operator. A comparison with the other commonly applied nonlinear algorithms, particularly the diffusion map, is provided. Moreover, we discuss the relationship between two common nearest neighbor search schemes and the relationship of LLE with the locally linear regression. At last, we consider the case when the point cloud is sampled from a manifold with boundary.

We show that if the regularization is chosen correctly, LLE algorithm asymptotically recovers a linear second order differential operator with “free” boundary condition. Such operator coincides with Laplace-Beltrami operator in the interior of the manifold. We further modify LLE algorithm to the Dirichlet Graph Laplacian algorithm which can be used to recover the Laplace-Beltrami operator of the manifold with Dirichlet boundary condition.

A copy of the thesis can be found here: Nan Wu’s thesis

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