Title: Vector diffusion maps and random matrices with random blocks

Abstract: Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps. It was previously shown by Belkin and Niyogi in 2007 that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many uniformly sampled data points. Recently, we introduced Vector Diffusion Maps (VDM) and showed that the Connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In the first part of the talk, we will present a unified framework for approximating other Connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many random samples. Our results for spectral convergence also hold in the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.

Motivated by this technique and the inevitable noise in real data, we further study matrices that are akin to the ones appearing in the null case of VDM, i.e the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the VDM algorithm - whether there is signal or not. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good.

The algorithm is directly applied to the class-averging algorithm in the cryo-EM problem and the result will be demonstrated. If time permits, further medical imaging applications will be discussed.