Mathematics Colloquia and Seminars

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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, and hence generalize the work by Belkin and Niyogi in 2007. We will show how VDM is applied to the ptychography problem. Motivated by this technique and the inevitable noise in real data, In the second part of the talk, 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 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 with dependent ``random strip structure''. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good. This is a joint work with Amit Singer, Noureddine El Karoui, Stefano Marchesini, etc.