On Monotone Embedding in Information Geometry
A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geo...
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MDPI AG
2015-06-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/17/7/4485 |
| _version_ | 1828283625743843328 |
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| author | Jun Zhang |
| author_facet | Jun Zhang |
| author_sort | Jun Zhang |
| collection | DOAJ |
| description | A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations. |
| first_indexed | 2024-04-13T08:48:09Z |
| format | Article |
| id | doaj.art-cd9c744fcfa04f73b0f95097f93e38f7 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-13T08:48:09Z |
| publishDate | 2015-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-cd9c744fcfa04f73b0f95097f93e38f72022-12-22T02:53:36ZengMDPI AGEntropy1099-43002015-06-011774485449910.3390/e17074485e17074485On Monotone Embedding in Information GeometryJun Zhang0Department of Psychology and Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USAA paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations.http://www.mdpi.com/1099-4300/17/7/4485α-embeddingmonotone embeddingconjugate embeddinggeneralized Fisher–Rao metricAmari–Chentsov tensordeformed logarithmrepresentation duality(ρ,τ)-geometry |
| spellingShingle | Jun Zhang On Monotone Embedding in Information Geometry Entropy α-embedding monotone embedding conjugate embedding generalized Fisher–Rao metric Amari–Chentsov tensor deformed logarithm representation duality (ρ,τ)-geometry |
| title | On Monotone Embedding in Information Geometry |
| title_full | On Monotone Embedding in Information Geometry |
| title_fullStr | On Monotone Embedding in Information Geometry |
| title_full_unstemmed | On Monotone Embedding in Information Geometry |
| title_short | On Monotone Embedding in Information Geometry |
| title_sort | on monotone embedding in information geometry |
| topic | α-embedding monotone embedding conjugate embedding generalized Fisher–Rao metric Amari–Chentsov tensor deformed logarithm representation duality (ρ,τ)-geometry |
| url | http://www.mdpi.com/1099-4300/17/7/4485 |
| work_keys_str_mv | AT junzhang onmonotoneembeddingininformationgeometry |