Back to Departmental Colloquium: Fall 2015
Departmental Colloquium
Date: Thursday, Oct 29, 2015
Time: 4:00PM
Location: LCB 219
Dejan Slepcev
Carnegie Mellon University
Title |
Variational problems on graphs and their continuum limits |
Abstract |
We discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying “ground-truth” measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and classification, can be posed as minimizing functionals on such graphs. We consider functionals involving graph cuts and their limits as the number of data points goes to infinity. In particular we establish under what conditions the minimizers of discrete problems have a well defined continuum limit, and characterize the limit. The question is considered using the Gamma convergence. The Gamma limit, and associated compactness property, are considered with respect to a topology which uses optimal transportation to suitably compare functions defined on graphs with functions defined with respect to the continuum ground-truth measure. The talk is primarily based on joint works with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Thomas Laurent, and James von Brecht. |
Commutative Algebra
Combinatorics
Topology