Spectral Clustering and Multidimensional Scaling: A Unified View

Abstract

Spectral clustering is a procedure aimed at partitionning a weighted graph into minimally interacting components. The resulting eigen-structure is determined by a reversible Markov chain, or equivalently by a symmetric transition matrix F. On the other hand, multidimensional scaling procedures (and factorial correspondence analysis in particular) consist in the spectral decomposition of a kernel matrix K. This paper shows how F and K can be related to each other through a linear or even non-linear transformation leaving the eigen-vectors invariant. As illustrated by examples, this circumstance permits to define a transition matrix from a similarity matrix between n objects, to define Euclidean distances between the vertices of a weighted graph, and to elucidate the “flow-induced” nature of spatial auto-covariances.