Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a Rényi Information Measure
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909327525150720 |
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| author | Tsai, Chung-En Wang, Guan-Ren Cheng, Hao-Chung Li, Yen-Huan |
| author_facet | Tsai, Chung-En Wang, Guan-Ren Cheng, Hao-Chung Li, Yen-Huan |
| contents | Consider the problems of computing the Augustin information and a Rényi information measure of statistical independence, previously explored by Lapidoth and Pfister (IEEE Information Theory Workshop, 2018) and Tomamichel and Hayashi (IEEE Trans. Inf. Theory, 64(2):1064--1082, 2018). Both quantities are defined as solutions to optimization problems and lack closed-form expressions. This paper analyzes two iterative algorithms: Augustin's fixed-point iteration for computing the Augustin information, and the algorithm by Kamatsuka et al. (arXiv:2404.10950) for the Rényi information measure. Previously, it was only known that these algorithms converge asymptotically. We establish the linear convergence of Augustin's algorithm for the Augustin information of order $α\in (1/2, 1) \cup (1, 3/2)$ and Kamatsuka et al.'s algorithm for the Rényi information measure of order $α\in [1/2, 1) \cup (1, \infty)$, using Hilbert's projective metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02640 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a Rényi Information Measure Tsai, Chung-En Wang, Guan-Ren Cheng, Hao-Chung Li, Yen-Huan Optimization and Control Information Theory Consider the problems of computing the Augustin information and a Rényi information measure of statistical independence, previously explored by Lapidoth and Pfister (IEEE Information Theory Workshop, 2018) and Tomamichel and Hayashi (IEEE Trans. Inf. Theory, 64(2):1064--1082, 2018). Both quantities are defined as solutions to optimization problems and lack closed-form expressions. This paper analyzes two iterative algorithms: Augustin's fixed-point iteration for computing the Augustin information, and the algorithm by Kamatsuka et al. (arXiv:2404.10950) for the Rényi information measure. Previously, it was only known that these algorithms converge asymptotically. We establish the linear convergence of Augustin's algorithm for the Augustin information of order $α\in (1/2, 1) \cup (1, 3/2)$ and Kamatsuka et al.'s algorithm for the Rényi information measure of order $α\in [1/2, 1) \cup (1, \infty)$, using Hilbert's projective metric. |
| title | Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a Rényi Information Measure |
| topic | Optimization and Control Information Theory |
| url | https://arxiv.org/abs/2409.02640 |