Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula
It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this...
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Institute of Electrical and Electronics Engineers (IEEE)
2020
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| Online Access: | https://hdl.handle.net/1721.1/124314 |
| _version_ | 1826205761647149056 |
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| author | Yang, Wei Collins, Austin Daniel Durisi, Giuseppe Polyanskiy, Yury Poor, H. Vincent |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Yang, Wei Collins, Austin Daniel Durisi, Giuseppe Polyanskiy, Yury Poor, H. Vincent |
| author_sort | Yang, Wei |
| collection | MIT |
| description | It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finite-blocklength channel-coding converse bound by Polyanskiy and Verdú, which involves the ratio of two Neyman-Pearson $\beta $ functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson $\beta $ functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verdú's wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver. |
| first_indexed | 2024-09-23T13:18:36Z |
| format | Article |
| id | mit-1721.1/124314 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:18:36Z |
| publishDate | 2020 |
| publisher | Institute of Electrical and Electronics Engineers (IEEE) |
| record_format | dspace |
| spelling | mit-1721.1/1243142022-09-28T13:19:10Z Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula Yang, Wei Collins, Austin Daniel Durisi, Giuseppe Polyanskiy, Yury Poor, H. Vincent Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finite-blocklength channel-coding converse bound by Polyanskiy and Verdú, which involves the ratio of two Neyman-Pearson $\beta $ functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson $\beta $ functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verdú's wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver. National Science Foundation (10.13039/100000001) Swedish Research Council Center for Science of Information 2020-03-25T14:31:20Z 2020-03-25T14:31:20Z 2018-05 2019-07-01T17:44:24Z Article http://purl.org/eprint/type/JournalArticle 0018-9448 1557-9654 https://hdl.handle.net/1721.1/124314 Yang, Wei et al. "Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula." IEEE Transactions on Information Theory 64, 9 (September 2018): 6236 - 6256 © 1963-2012 IEEE. en http://dx.doi.org/10.1109/tit.2018.2837104 IEEE Transactions on Information Theory Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Electrical and Electronics Engineers (IEEE) arXiv |
| spellingShingle | Yang, Wei Collins, Austin Daniel Durisi, Giuseppe Polyanskiy, Yury Poor, H. Vincent Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula |
| title | Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula |
| title_full | Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula |
| title_fullStr | Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula |
| title_full_unstemmed | Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula |
| title_short | Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula |
| title_sort | beta beta bounds finite blocklength analog of the golden formula |
| url | https://hdl.handle.net/1721.1/124314 |
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