Information inequalities and finite groups : an overview
An entropic vector is a 2n - 1 dimensional vector collecting all the possible joint entropies of n discrete jointly distributed random variables. The region Γ*n of entropic vectors plays a significant role in network information theory, since it is known that the capacity of large classes of network...
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| Format: | Conference Paper |
| Language: | English |
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2015
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| Online Access: | https://hdl.handle.net/10356/104953 http://hdl.handle.net/10220/24987 http://ieeexplore.ieee.org.ezlibproxy1.ntu.edu.sg/xpl/articleDetails.jsp?tp=&arnumber=6979933&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6979933 |
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| author | Markin, Nadya Oggier, Frédérique |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Markin, Nadya Oggier, Frédérique |
| author_sort | Markin, Nadya |
| collection | NTU |
| description | An entropic vector is a 2n - 1 dimensional vector collecting all the possible joint entropies of n discrete jointly distributed random variables. The region Γ*n of entropic vectors plays a significant role in network information theory, since it is known that the capacity of large classes of networks can be computed by optimising a linear function over Γ*n, under linear constraints, where n is the number of variables involved in a given network. However so far only Γ*2 and Γ*3 are known. One approach to study Γ*n is to identify smaller regions that serve as inner bounds, and to look for entropic vectors violating inequalities characterising these regions. For example for n = 4, the inequality of interest is the so-called Ingleton inequality. We give an overview of recent work studying entropic vectors using a group theoretic approach. We recall Chan's technique of constructing random variables from groups and the corresponding notion of (abelian) group representable entropic vectors. We review different works on groups yielding violations of linear rank inequalities, and discuss the classification of finite groups based on the entropic vector that they yield. |
| first_indexed | 2025-02-19T04:01:26Z |
| format | Conference Paper |
| id | ntu-10356/104953 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2025-02-19T04:01:26Z |
| publishDate | 2015 |
| record_format | dspace |
| spelling | ntu-10356/1049532023-02-28T19:18:07Z Information inequalities and finite groups : an overview Markin, Nadya Oggier, Frédérique School of Physical and Mathematical Sciences 2014 International Symposium on Information Theory and its Applications (ISITA) DRNTU::Science::Mathematics An entropic vector is a 2n - 1 dimensional vector collecting all the possible joint entropies of n discrete jointly distributed random variables. The region Γ*n of entropic vectors plays a significant role in network information theory, since it is known that the capacity of large classes of networks can be computed by optimising a linear function over Γ*n, under linear constraints, where n is the number of variables involved in a given network. However so far only Γ*2 and Γ*3 are known. One approach to study Γ*n is to identify smaller regions that serve as inner bounds, and to look for entropic vectors violating inequalities characterising these regions. For example for n = 4, the inequality of interest is the so-called Ingleton inequality. We give an overview of recent work studying entropic vectors using a group theoretic approach. We recall Chan's technique of constructing random variables from groups and the corresponding notion of (abelian) group representable entropic vectors. We review different works on groups yielding violations of linear rank inequalities, and discuss the classification of finite groups based on the entropic vector that they yield. Accepted version 2015-02-02T01:45:02Z 2019-12-06T21:43:24Z 2015-02-02T01:45:02Z 2019-12-06T21:43:24Z 2014 2014 Conference Paper Markin, N., & Oggier, F. (2014). Information inequalities and finite groups : an overview. 2014 International Symposium on Information Theory and its Applications (ISITA). https://hdl.handle.net/10356/104953 http://hdl.handle.net/10220/24987 http://ieeexplore.ieee.org.ezlibproxy1.ntu.edu.sg/xpl/articleDetails.jsp?tp=&arnumber=6979933&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6979933 182299 en © 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at: [http://ieeexplore.ieee.org.ezlibproxy1.ntu.edu.sg/xpl/articleDetails.jsp?tp=&arnumber=6979933&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6979933]. application/pdf |
| spellingShingle | DRNTU::Science::Mathematics Markin, Nadya Oggier, Frédérique Information inequalities and finite groups : an overview |
| title | Information inequalities and finite groups : an overview |
| title_full | Information inequalities and finite groups : an overview |
| title_fullStr | Information inequalities and finite groups : an overview |
| title_full_unstemmed | Information inequalities and finite groups : an overview |
| title_short | Information inequalities and finite groups : an overview |
| title_sort | information inequalities and finite groups an overview |
| topic | DRNTU::Science::Mathematics |
| url | https://hdl.handle.net/10356/104953 http://hdl.handle.net/10220/24987 http://ieeexplore.ieee.org.ezlibproxy1.ntu.edu.sg/xpl/articleDetails.jsp?tp=&arnumber=6979933&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6979933 |
| work_keys_str_mv | AT markinnadya informationinequalitiesandfinitegroupsanoverview AT oggierfrederique informationinequalitiesandfinitegroupsanoverview |