Variational Methods and PDEs on Graphs with Applications in Data Processing and Machine Learning; Dr. Daniel Tenbrinck (WWU Münster)

ABSTRACT: Graph-based methods have emerged as a promising tool for many applications in machine learning and data processing. One key feature of these methods is the possibility to incorporate nonlocal relationships in the data rather than using only local neighborhoods. The recent trend in the literature is to translate well-studied variational problems and PDEs to the graph setting and overcome hereby drawbacks of classical approaches.
In this talk we give a short introduction to the concept of partial difference equations on graphs and show that classical numerical discretization schemes can be embedded in a graph setting and thus be interpreted as special cases in a more general framework. To give an example we discuss a family of graph p- and ∞-Laplacians. We analyze the corresponding PDEs involving these operators which enable us to perform important processing steps, such as diffusion-based filtering and interpolation of data.
Finally, we demonstrate the advantages of graph-based methods for different tasks in image and point cloud processing, i.e., filtering, segmentation, inpainting, and machine learning.

This is a joint joint work with A. Elmoataz, Université de Caen, France.