Title: Variational problems on graphs and their continuum limits
Speaker: Dejan Slepcev (CMU)
Abstract: The talk discusses variational problems arising in machine learning and their consistency as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure on a Euclidean domain. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. I will discuss when is the graph cut, and more generally, total variation, on such graphs a good approximation of the perimeter (total variation) in the continuum setting. The question is considered in the setting of $\Gamma$-convergence. The $\Gamma$-limit, and associated compactness property, are considered with respect to a topology which uses optimal transportation to suitably compare $L^p$ functions defined with respect to different measures. Applications to consistency of spectral clustering will also be discussed. The talk is primarily based on joint work with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Thomas Laurent, and James von Brecht.
Seminar URL: https://research.math.osu.edu/tgda/tgda-seminar.html
