Graph connection Laplacian, principal bundle structure and image processing

Abstract: Spectral methods like Diffusion Maps and Laplacian Eigenmaps in data analysis are based on eigenvectors and eigenvalues of graph Laplacians. Recently, we introduced the graph connection Laplacian and its derivation vector diffusion maps (VDM) and vector diffusion distance (VDD), and we showed that under the manifold assumption, asymptotically the graph connection Laplacian approximates the connection Laplacian of the tangent bundle of the manifold and VDD approximates the geodesic distance. We will present a unified framework for approximating different kinds of connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of this generalized algorithm 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.

We will discuss the real applications of this framework in the class-averging algorithm in the cryo-EM problem based on VDD and the ptychography (a phase retrieval) problem based on the graph connection Laplacian. If time permits, the random matrix analysis of the noise impact on the VDM algorithm will be discussed, which shows the robustness of the proposed algorithm to the inevitable noise we face in the real application. This is a joint work with Noureddine El Karoui, Stefano Marchesini, Amit Singer, etc.