Stability and Pattern formation in Nonlocal Interaction Models

Abstract. Geometric methods based on PDEs have revolutionized the field of im-age processing and image analysis. I will discuss recent work to develop these ideas for machine learning applications involving “big data”. The main idea is to pose variational problems involving graph cuts in terms of total variation minimization problems. We then develop both phase field and mean curvature methods to solve these problems quickly. I will introduce the notion of the Ginzburg-Landau func-tional on graphs and the related dynamic thresholding method. Unlike numerical methods for PDEs, the graph problems are able to exploit dramatic spectral trunca-tion of the graph Laplacian, sometimes with a tiny fraction of the eigenfunctions. I will show examples from semi-supervised learning, nonlocal means image processing, and modularity optimization for unsupervised learning and community detection in networks