[Paper Review] Spectral clustering of Markov chain transition matrices with complex eigenvalues

Anna-Simone Frank, Alexander Sikorski+1 more|arXiv (Cornell University)|Jun 29, 2022

Bayesian Methods and Mixture Models4 citations

TL;DR

This paper introduces cPCCA+, a spectral clustering method that extends the Robust Perron Cluster Analysis (PCCA+) to Markov chains with complex eigenvalues by decomposing complex conjugate eigenvectors into real and imaginary components, enabling efficient and accurate coarse-graining of non-reversible stochastic processes such as circular chains and gene-regulatory networks. The method achieves comparable accuracy to GenPCCA but with significantly faster runtime in MATLAB due to avoidance of computationally expensive Schur decomposition.

ABSTRACT

The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GenPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains,and demonstrate its efficiency compared to GenPCCA. The experiments are performed in the Matlab programming language and codes are provided.

Motivation & Objective

• To extend the applicability of PCCA+ beyond reversible Markov chains with real eigenvalues to non-reversible chains with complex eigenvalues.
• To address the computational inefficiency of GenPCCA in MATLAB, particularly due to direct Schur decomposition and sorting routines.
• To provide a theoretically sound and computationally efficient alternative to GenPCCA for spectral clustering of complex eigen-decomposed transition matrices.
• To demonstrate the method’s effectiveness on real-world examples, including circular transition matrices and stochastic gene-regulatory networks.

Proposed method

• Replace complex conjugate eigenvector pairs with their real and imaginary components to form a real subspace representation suitable for cluster analysis.
• Apply the standard PCCA+ clustering framework to the real-valued subspace, preserving the invariant subspace condition required for metastable state identification.
• Use eigen-decomposition of the transition matrix P to extract dominant eigenvalues and eigenvectors, including complex conjugate pairs.
• Ensure the resulting coarse-grained transition matrix P_c maintains slow inter-macrostate transitions and fast intra-macrostate transitions, as required by PCCA+.
• Implement the algorithm in MATLAB and compare performance with GenPCCA using Schur decomposition and sorting routines.
• Validate the method on synthetic circular chains and a stochastic gene-regulatory network model, measuring runtime and accuracy.

Experimental results

Research questions

• RQ1 Can PCCA+ be extended to handle Markov chains with complex eigenvalues through a real-valued representation of the eigenspace?
• RQ2 Does the real subspace formed by separating real and imaginary parts of complex eigenvectors preserve the invariant subspace condition required for spectral clustering?
• RQ3 How does the computational performance of cPCCA+ compare to GenPCCA in MATLAB for increasing state space sizes?
• RQ4 Can cPCCA+ accurately identify metastable states in non-reversible systems such as circular chains and gene-regulatory networks?

Key findings

• cPCCA+ successfully extends PCCA+ to non-reversible Markov chains with complex eigenvalues by transforming complex eigenvectors into real components without loss of accuracy.
• The method maintains the invariant subspace condition when using real and imaginary parts of complex eigenvectors, ensuring valid cluster assignments.
• cPCCA+ achieves comparable clustering accuracy to GenPCCA on both circular transition matrices and gene-regulatory network models.
• cPCCA+ demonstrates significantly faster runtime than GenPCCA in MATLAB, especially for larger state spaces, due to avoidance of direct Schur decomposition and sorting.
• For the gene-regulatory network example, the runtime difference between cPCCA+ and GenPCCA grows quadratically with state space size, with cPCCA+ consistently outperforming GenPCCA.
• The method is robust to small perturbations in matrix elements, as such matrices typically admit eigen-decompositions, making the approach practically viable.