1. Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a R\'{e}nyi Information Measure
- Author
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Tsai, Chung-En, Wang, Guan-Ren, Cheng, Hao-Chung, and Li, Yen-Huan
- Subjects
Mathematics - Optimization and Control ,Computer Science - Information Theory - Abstract
Consider the problems of computing the Augustin information and a R\'{e}nyi information measure of statistical independence, previously explored by Lapidoth and Pfister (\textit{IEEE Information Theory Workshop}, 2018) and Tomamichel and Hayashi (\textit{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 R\'{e}nyi 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 $\alpha \in (1/2, 1) \cup (1, 3/2)$ and Kamatsuka et al.'s algorithm for the R\'{e}nyi information measure of order $\alpha \in [1/2, 1) \cup (1, \infty)$, using Hilbert's projective metric., Comment: 15 pages
- Published
- 2024