Graph Laplacians and Markov processes are intimately connected and ubiquitous in the study of graph structures. They have led to significant advances in a class of geometric inverse problems known as “manifold learning”, wherein one wishes to learn the geometry of a Riemannian submanifold from finite Euclidean point samples. The data gives rise to the geometry-encoding neighbourhood graphs. Present-day techniques are dominated primarily by the low spectral resolution of the graph Laplacians, while finer aspects of the underlying geometry, such as the geodesic flow, are observed only in the high spectral regime. We establish a data-driven* uncertainty principle* that dictates the scaling of the wavelength h, with respect to the density of samples, at which graph Laplacians are approximately h-pseudodifferential operators. This sets the stage for a semiclassical approach to the high-frequency analysis of wave dynamics on weighted graphs. We thus establish a discrete version of Egorov’s theorem and achieve convergence rates for the recovery of geodesics on the underlying manifolds through quantum dynamics on the approximating graphs. I will briefly show some applications to real-world datasets and connections to approximation of other pseudodifferential operators on manifolds.

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# Wave dynamics and semiclassical analysis: from graphs to manifolds

Akshat Kumar (Instituto de Telecomunicações, Lisboa)

Tuesday, November 14, 2023 - 1:30pm to 3:30pm

PDL C-401