# A Theory of Spectral Clustering

# A Theory of Spectral Clustering

Date

21 Nov 2018

h. 15.30

h. 15.30

Location

Aula Màster del Campus Nord,

UPC

UPC

**Abstract**

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 dimension-reduction 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.

Date

21 Nov 2018

h. 15.30

h. 15.30

Location

Aula Màster del Campus Nord,

UPC

UPC

**Abstract**

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 dimension-reduction 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.

Luca Trevisan, University of Berkeley (USA)