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Applied and Computational Math Seminar

The geometry of data through the lens of homogenization.

Raghav Venkatraman (Courant Institute)

Location:  Hill 705
Date & time: Wednesday, 28 February 2024 at 11:00AM - 12:00PM

The ``manifold hypothesis'' posits that many high dimensional data sets that occur in the real world actually are actually scattered on or around a much lower dimensional manifold embedded in the high dimensional space. Estimating attributes of this "ground truth" manifold from finitely many samples (point cloud) is a problem of statistical inference. Given such a point cloud that is modelled as an independent and identically distributed (i.i.d) sample from a (nice) density on a closed manifold, over the past decade there is a body of literature which considers the question: forming a random geometric (weighted) graph on the point cloud (by, for example, joining points that are within a threshold distance by a weighted edge) how well can one estimate the spectrum (eigenvalues, eigenfunctions) of the (weighted) Laplace-Beltrami operator on the ground truth manifold, from that of the graph laplacian associated with the random geometric graph?

 

After introducing the problem, we will show how this question is one of ``stochastic homogenization'', a traditionally well-studied theme in partial differential equations originating in the theory of composite materials. Warming up with results that are new even for the classical "periodic" homogenization problem, we will describe how one can obtain optimal convergence rates for the spectrum of the graph laplacian using tools from the recent theory of quantitative stochastic homogenization. Briefly: borrowing tools from percolation theory, the argument proceeds by ``coarse-graining'' the random geometry in the problem to large scales, where the environment "appears Euclidean". Then, one adapts  arguments from the more recent quantitative theory of homogenization. 

 

This talk represents joint work with Scott Armstrong (Courant). 

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