We show how to build hierarchical, reduced-rank representation for large stochastic matrices and use this representation to design an efficient algorithm forcomputing the largest eigenvalues, and the corresponding eigenvectors. In particular, the eigen problem is first solved at the coarsest levelof the representation. The approximate eigen solution is then interpolated over successive levels of the hierarchy. A small number of power iterations are employed at each stage to correct the eigen solution. The typical speedups obtained by a Matlab implementation of our fast eigensolver over a standard sparse matrix eigensolver [13] are at least a factor of ten for large image sizes. The hierarchical representation has proven to be effective in a min-cut based segmentation algorithm that we proposed recently [8].

Using a Markov chain perspective of spectral clustering we present an algorithm to automatically find the number of stable clusters in a dataset. The Markov chain's behaviour is characterized by the spectral properties of the matrix of transition probabilities, from which we derive eigenflows along with their halflives. An eigenflow describes the flow of probability mass due to the Markov chain, and it is characterized by its eigenvalue, or equivalently, by the halflife of its decay as the Markov chain is iterated. A ideal stable cluster is one with zero eigenflow and infinite half-life. The key insight in this paper is that bottlenecks between weakly coupled clusters can be identified by computing the sensitivity of the eigenflow's halflife to variations in the edge weights.

Using a Markov chain perspective of spectral clustering we present an algorithm to automatically find the number of stable clusters in a dataset. The Markov chain's behaviour is characterized by the spectral properties of the matrix of transition probabilities, from which we derive eigenflows along with their halflives. An eigenflow describes the flow of probability massdue to the Markov chain, and it is characterized by its eigenvalue, orequivalently, by the halflife of its decay as the Markov chain is iterated. A ideal stable cluster is one with zero eigenflow and infinite half-life.The key insight in this paper is that bottlenecks between weakly coupled clusters can be identified by computing the sensitivity of the eigenflow's halflife to variations in the edge weights. We propose a novel EIGENCUTS algorithm to perform clustering that removes these identified bottlenecks in an iterative fashion.