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Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification

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Abstract

Given a row-stochastic matrix describing pairwise similarities between data objects, spectral clustering makes use of the eigenvectors of this matrix to perform dimensionality reduction for clustering in fewer dimensions. One example from this class of algorithms is the Robust Perron Cluster Analysis (PCCA+), which delivers a fuzzy clustering. Originally developed for clustering the state space of Markov chains, the method became popular as a versatile tool for general data classification problems. The robustness of PCCA+, however, cannot be explained by previous perturbation results, because the matrices in typical applications do not comply with the two main requirements: reversibility and nearly decomposability. We therefore demonstrate in this paper that PCCA+ always delivers an optimal fuzzy clustering for nearly uncoupled, not necessarily reversible, Markov chains with transition states.

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Notes

  1. The Perron eigenvalue is the unique largest real eigenvalue of a real square matrix with positive entries.

  2. Unfortunately, the statement that \(R_i\) equals zero as given in Deuflhard and Weber (2005) turned out to be wrong, see Kube and Deuflhard (2006). Nevertheless, the result explains the robustness of the eigenvectors under perturbations.

  3. http://www.gnu.org/software/glpk.

  4. http://www.zib.de/de/numerik/software/newtonlib.html.

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Authors and Affiliations

  1. Zuse Institute Berlin (ZIB), Takustraße 7, 14195 , Berlin, Germany

    Susanna Röblitz & Marcus Weber

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  1. Susanna Röblitz
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  2. Marcus Weber
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Correspondence to Susanna Röblitz.

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Röblitz, S., Weber, M. Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification. Adv Data Anal Classif 7, 147–179 (2013). https://doi.org/10.1007/s11634-013-0134-6

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