Wave dynamics and semiclassical analysis: from graphs to manifolds

The HADES seminar on Tuesday, December 5th, will be at 3:30pm in Room 748 (not in 740 this week!).

Speaker: Akshat Kumar

Abstract: 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 for neighbourhood graphs 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 show examples on samples of model manifolds and briefly discuss some applications to real-world datasets.

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