Rényi Cross-Entropy Measures for Common Distributions and Processes with Memory
Abstract
:1. Introduction
2. Rényi and Natural Rényi Differential Cross-Entropies for Distributions from the Exponential Family
3. Tables of Rényi and Natural Rényi Differential Cross-Entropies
4. Rényi and Natural Rényi Differential Cross-Entropy Rates for Stationary Gaussian Processes
5. Rényi and Natural Rényi Cross-Entropy Rates for Markov Sources
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Name | Parameters () | Support | |
---|---|---|---|
Beta | (, ) | ||
(, ) | |||
() | |||
() | |||
Exponential | () | ||
Gamma | (, ) | ||
(,) | |||
(, ) | |||
Half-Normal | () | ||
Gumbel | (, ) | ||
Pareto | (, ) | ||
Maxwell Boltzmann | () | ||
Rayleigh | () | ||
Laplace | (, ) |
- is the Beta function.
- is the Gamma function.
References
- Rényi, A. On measures of entropy and information. Fourth Berkeley Symp. Math. Stat. Probab. 1961, 1, 547–561. [Google Scholar]
- Verdú, S. α-Mutual Information. In Proceedings of the 2015 Information Theory and Applications Workshop (ITA), San Diego, CA, USA, 1–6 February 2015; pp. 1–6. [Google Scholar]
- Sarraf, A.; Nie, Y. RGAN: Rényi generative adversarial network. SN Comput. Sci. 2021, 2, 17. [Google Scholar] [CrossRef]
- Valverde-Albacete, F.J.; Peláez-Moreno, C. The Case for shifting the Rényi entropy. Entropy 2019, 21, 46. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Bhatia, H.; Paul, W.; Alajaji, F.; Gharesifard, B.; Burlina, P. Least kth-Order and Rényi generative adversarial networks. Neural Comput. 2021, 33, 2473–2510. [Google Scholar] [CrossRef] [PubMed]
- Gil, M.; Alajaji, F.; Linder, T. Rényi divergence measures for commonly used univariate continuous distributions. Inf. Sci. 2013, 249, 124–131. [Google Scholar] [CrossRef]
- Song, K.S. Rényi information, loglikelihood and an intrinsic distribution measure. J. Statist. Plann. Inference 2001, 93, 51–69. [Google Scholar] [CrossRef]
- Thierrin, F.C.; Alajaji, F.; Linder, T. On the Rényi cross-entropy. In Proceedings of the 17th Canadian Workshop on Information Theory, Ottawa, ON, Canada, 5–8 June 2022; pp. 1–5. [Google Scholar] [CrossRef]
- Goodfellow, I.; Pouget-Abadie, J.; Mirza, M.; Xu, B.; Warde-Farley, D.; Ozair, S.; Courville, A.; Bengio, Y. Generative adversarial nets. Adv. Neural Inf. Process. Syst. 2014, 27, 2672–2680. [Google Scholar]
- Bhatia, H.; Paul, W.; Alajaji, F.; Gharesifard, B.; Burlina, P. Rényi generative adversarial Networks. arXiv 2020, arXiv:2006.02479v1. [Google Scholar]
- Kluza, P.A. On Jensen-Rényi and Jeffreys-Rényi type f-divergences induced by convex functions. Phys. Stat. Mech. Its Appl. 2019, 548, 122527. [Google Scholar] [CrossRef]
- Lin, J. Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory 1991, 31, 145–151. [Google Scholar] [CrossRef] [Green Version]
- Pantazis, Y.; Paul, D.; Fasoulakis, M.; Stylianou, Y.; Katsoulakis, M. Cumulant GAN. arXiv 2020, arXiv:2006.06625. [Google Scholar] [CrossRef]
- Kurri, G.R.; Sypherd, T.; Sankar, L. Realizing GANs via a tunable loss function. In Proceedings of the IEEE Information Theory Workshop (ITW), Kanazawa, Japan, 17–21 October 2021; pp. 1–6. [Google Scholar]
- Kurri, G.R.; Welfert, M.; Sypherd, T.; Sankar, L. α-GAN: Convergence and estimation guarantees. In Proceedings of the IEEE International Symposium on Information Theory (ISIT), Espoo, Finland, 26 June–1 July 2022; pp. 312–317. [Google Scholar]
- Rached, Z.; Alajaji, F.; Campbell, L.L. Rényi’s divergence and entropy rates for finite alphabet Markov sources. IEEE Trans. Inf. Theory 2001, 47, 1553–1561. [Google Scholar] [CrossRef]
- Seneta, E. Non-Negative Matrices and Markov Chains; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Gallager, R.G. Discrete Stochastic Processes; Springer: Berlin/Heidelberg, Germany, 1996; pp. 31–55. [Google Scholar]
- Csiszar, I. Eine Informationstheoretische Ungleichung und ihre Anwendung auf den Bewis der Ergodizitat on Markhoffschen Ketten. Publ. Math. Inst. Hung. Acad. Sci. Ser. A 2006, 8, 85–108. [Google Scholar]
- Csiszár, I. Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung. 1967, 2, 299–318. [Google Scholar]
- Ali, S.M.; Silvey, S.D. A general class of coefficients of divergence of one distribution from another. J. R. Stat. Society. Ser. B (Methodol.) 1966, 28, 131–142. [Google Scholar] [CrossRef]
- Liese, F.; Vajda, I. On divergences and informations in statistics and information theory. IEEE Trans. Inf. Theory 2006, 52, 4394–4412. [Google Scholar] [CrossRef]
Name | |
---|---|
Beta | |
, | |
, | |
, | |
, | |
, | |
, | |
Exponential | |
, | |
Gamma | |
, | |
, | |
, | |
, | |
, | |
Half-Normal | |
, | |
, | |
, | |
, | |
Rayleigh | |
, |
Name | |
---|---|
Beta | |
, | |
, | |
, | |
, | |
, | |
, | |
Exponential | |
, | |
Gamma | |
, , | |
, | |
, | |
, | |
Half-Normal | |
, | |
, | |
Maxwell Boltzmann | |
, | |
, | |
Rayleigh | |
, |
Information Measure | Rate | Constraint |
---|---|---|
Information Measure | Rate |
---|---|
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Thierrin, F.C.; Alajaji, F.; Linder, T. Rényi Cross-Entropy Measures for Common Distributions and Processes with Memory. Entropy 2022, 24, 1417. https://doi.org/10.3390/e24101417
Thierrin FC, Alajaji F, Linder T. Rényi Cross-Entropy Measures for Common Distributions and Processes with Memory. Entropy. 2022; 24(10):1417. https://doi.org/10.3390/e24101417
Chicago/Turabian StyleThierrin, Ferenc Cole, Fady Alajaji, and Tamás Linder. 2022. "Rényi Cross-Entropy Measures for Common Distributions and Processes with Memory" Entropy 24, no. 10: 1417. https://doi.org/10.3390/e24101417
APA StyleThierrin, F. C., Alajaji, F., & Linder, T. (2022). Rényi Cross-Entropy Measures for Common Distributions and Processes with Memory. Entropy, 24(10), 1417. https://doi.org/10.3390/e24101417