BIDSA SEMINAR: A Theory of Spectral Clustering

Luca Trevisan, University of California, Berkeley

December 18, 2017

12:30pm, Bocconi University

Via Rӧntgen n.1, Rm 3 e4 sr03

Spectral clustering algorithms find clusters in a given network by exploiting properties of the eigenvectors of matrices associated with the network. As a first step, one computes a spectral embedding, that is a mapping of nodes to points in a low-dimensional real space; then one uses geometric clustering algorithms such as k-means to cluster the points corresponding to the nodes.Such algorithms work so well that, in certain applications unrelated to network analysis, such as image segmentation, it is useful to associate a network to the data, and then apply spectral clustering to the network. In addition to its application to clustering, spectral embeddings are a valuable tool for dimensionreduction and data visualization. The performance of spectral clustering algorithms has been justified rigorously when applied to networks coming from certain probabilistic generative models. A more recent development, which is the focus of this lecture, is a worst-case analysis of spectral clustering, showing that, for every graph that exhibits a certain cluster structure, such structure can be found by geometric algorithms applied to a spectral embedding. Such results generalize the graph Cheeger's inequality (a classical result in spectral graph theory), and they have additional applications in computational complexity theory and in pure mathematics.