Generalizing the Motzkin-Straus Theorem to Edge-Weighted Graphs, with Applications to Image Segmentation

The Motzkin-Straus theorem is a remarkable result from graph theory that has recently found various applications in computer vision and pattern recognition. Given an unweighted undirected graph G with adjacency matrix A, it establishes a connection between the local/global solutions of the following quadratic program: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ maximize x^T A x / 2 $$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$subject to e^T x=1, x \in \mathbb{R}_+^n $$\end{document} where e = (1,...,1)T, and the maximal/maximum cliques of G. Given an edge-weighted undirected graph G and the corresponding weight matrix A, in this paper we address the following question: What kind of (combinatorial) structures of G are associated to the (continuous) local solutions of our quadratic program? We show that these structures correspond to a “weighted” generalization of maximal cliques, thereby providing a first step towards an edge-weighted generalization of the Motzkin-Straus theorem. Moreover, we show how these structures can be relevant in clustering as well as image segmentation problems. We present experimental results on real-world images which show the effectiveness of the proposed approach.