Computing the Nonnegative Low-Rank Leading Eigenmatrix and its Applications to Markov Grids and Metzler Operators
This work addresses computational challenges in analyzing Markov grids and Metzler operators, which are relevant for modeling population and epidemic dynamics, but it appears incremental as it builds on existing eigenpair approximation methods.
The paper tackles the problem of computing a nonnegative low-rank approximation of the rightmost eigenpair of linear matrix-valued operators, proposing an algorithm based on time integration of a differential system with nonnegative factorization. The result demonstrates effectiveness compared to standard approaches in applications like Markov grids and growth-diffusion operators.
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.