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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2204.00573 |
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| _version_ | 1866912143977218048 |
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| author | Parasnis, Rohit Franceschetti, Massimo Touri, Behrouz |
| author_facet | Parasnis, Rohit Franceschetti, Massimo Touri, Behrouz |
| contents | We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique. Our results apply to both discrete-time and continuous-time settings. We then discuss a few applications of our main results to non-Bayesian learning, distributed optimization, opinion dynamics, and averaging dynamics over random networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_00573 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains Parasnis, Rohit Franceschetti, Massimo Touri, Behrouz Optimization and Control We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique. Our results apply to both discrete-time and continuous-time settings. We then discuss a few applications of our main results to non-Bayesian learning, distributed optimization, opinion dynamics, and averaging dynamics over random networks. |
| title | A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2204.00573 |