WIAS Preprint No. 2891, (2021)

Coarse-graining and reconstruction for Markov matrices


  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2020 Mathematics Subject Classification

  • 60J10 15A42 39B62 05B20 60J28 15B52


  • Model-order reduction, stochastic matrix, Markov matrix, generalized Penrose--Moore inverse, coarse-graining and reconstruction, clustering, flux reconstruction, discrete functional inequalities, discrete Dirichlet forms, Poincaré-type constants




We present a coarse-graining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized Penrose-Moore inverse of the coarse-graining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we show, the method provides coarse-graining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarse-grain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincaré-type constants.

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