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Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a R\'{e}nyi Information Measure

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arxiv 2409.02640 v2 pith:KVJ4Q7OV submitted 2024-09-04 math.OC cs.ITmath.IT

Chung-En Tsai , Guan-Ren Wang , Hao-Chung Cheng , Yen-Huan Li This is my paper · ORCID
classification math.OCcs.ITmath.IT
keywords informationaugustinmeasurealgorithmcomputingalgorithmsalphaconvergence
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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 (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 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.

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Cited by 2 Pith papers

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  1. Alternating minimization for computing doubly minimized Petz Renyi mutual information

    quant-ph 2025-07 accept novelty 7.0 of 10

    Alternating minimization provably computes the doubly minimized Petz Rényi mutual information for all quantum states, with linear convergence for α∈(1,2] and O(1/n) convergence for α∈(1/2,1).

  2. A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean

    quant-ph 2025-02 conditional novelty 6.0 of 10

    A fixed-point iteration computes the Petz-Augustin mean with linear convergence in the Thompson metric for alpha > 1/2, giving the first non-asymptotic guarantees for this quantity and for the Petz capacity.

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