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Computing the Nonnegative Low-Rank Leading Eigenmatrix and its Applications to Markov Grids and Metzler Operators

Nicolas Gillis, Carmela Scalone

Abstract

We consider in this paper the problem of computing a nonnegative low-rank approximation of the rightmost eigenpair of a linear matrix-valued real operator. We propose an algorithm based on the time integration of a suitable differential system, whose solution is parametrized according to a nonnegative factorization. The conservation of the nonnegativity is theoretically motivated by the Perron-Frobenius theorem, while the computation of the rightmost eigenpair is motivated by two applications: (1) a new class of Markov chains, which we called Markov grids, whose transition matrices can be decomposed as the sum of Kronecker products, and (2) spatially structured systems in growth-diffusion operators arising for example in population and epidemic dynamics. Theoretical analysis and computational experiments show the effectiveness of the algorithm compared to standard approaches.

Computing the Nonnegative Low-Rank Leading Eigenmatrix and its Applications to Markov Grids and Metzler Operators

Abstract

We consider in this paper the problem of computing a nonnegative low-rank approximation of the rightmost eigenpair of a linear matrix-valued real operator. We propose an algorithm based on the time integration of a suitable differential system, whose solution is parametrized according to a nonnegative factorization. The conservation of the nonnegativity is theoretically motivated by the Perron-Frobenius theorem, while the computation of the rightmost eigenpair is motivated by two applications: (1) a new class of Markov chains, which we called Markov grids, whose transition matrices can be decomposed as the sum of Kronecker products, and (2) spatially structured systems in growth-diffusion operators arising for example in population and epidemic dynamics. Theoretical analysis and computational experiments show the effectiveness of the algorithm compared to standard approaches.
Paper Structure (26 sections, 2 theorems, 58 equations, 2 figures, 8 tables, 1 algorithm)

This paper contains 26 sections, 2 theorems, 58 equations, 2 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline and contribution of the paper
  3. Motivations: Markov grids and Meltzer operators
  4. Markov grids
  5. Interpretation
  6. Link with standard Markov chains
  7. Computational advantages
  8. Example: Tracking a moving object
  9. Adaptation of the model
  10. Low-rankness of solutions
  11. Nonnegativity of low-rank solutions
  12. Positive Dominant Modes in Spatially Structured Systems
  13. Algorithm using differential equations
  14. Matrix ODE for the low-rank rightmost eigenpair
  15. Taking nonnegativity into account
  16. ...and 11 more sections

Key Result

Theorem 1

Let $A$ and $B$ be irreducible transition matrices, and $\alpha \in \mathbb{R}^3_+$ be on the unit simplex with $\alpha_1 \neq 1$ and $\alpha_2 \neq 1$. Consider the Markov grid with transition Then the stationary state of this Markov grid , $X^*$, has rank-one, and is given by $X^* = \mu_A \mu_B^\top > 0$, where $\mu_A > 0$ and $\mu_B > 0$ are the stationary states of $A$ and $B$, respectively.

Figures (2)

  • Figure 1: Markov chain on a 3-by-3 grid.
  • Figure 2: Markov chain with 3 states.

Theorems & Definitions (10)

  • Example 1: Illustrative example
  • Definition 1: Markov grid
  • Remark 1: Kronecker factorization
  • Theorem 1
  • proof
  • Example 2
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof