Variational problems on random structures and their continuum limits

Abstract: We will 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 semi-supervised learning, can be posed as minimizing  functionals on such graphs. We consider functionals involving graph cuts, graph laplacians and their limits as the number of data points goes to infinity. We will discuss  the limits of functionals  when 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.

Biography: Dejan Slepcev is Professor of Mathematical Sciences at Carnegie Mellon University. He obtained his PhD in 2002 from University of Texas at Austin. His research interests include applied analysis, machine learning, and optimal transportation.