{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T14:56:59Z","timestamp":1763564219555,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T00:00:00Z","timestamp":1643241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"United States Air Force Office of Scientific Research","award":["AFOSR-FA9550-19-1-0213"],"award-info":[{"award-number":["AFOSR-FA9550-19-1-0213"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper systematically presents the \u03bb-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the \u03bb-deformed case: \u03bb-convexity, \u03bb-conjugation, \u03bb-biorthogonality, \u03bb-logarithmic divergence, \u03bb-exponential and \u03bb-mixture families, etc. In particular, \u03bb-deformation unifies Tsallis and R\u00e9nyi deformations by relating them to two manifestations of an identical \u03bb-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the \u03bb-exponential family, in turn, coincides with the \u03bb-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, \u03bb-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.<\/jats:p>","DOI":"10.3390\/e24020193","type":"journal-article","created":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T21:59:55Z","timestamp":1643320795000},"page":"193","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["\u03bb-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature"],"prefix":"10.3390","volume":"24","author":[{"given":"Jun","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Psychology, University of Michigan, Ann Arbor, MI 48109-1109, USA"},{"name":"Department of Statistics, University of Michigan, Ann Arbor, MI 48109-1109, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5254-7305","authenticated-orcid":false,"given":"Ting-Kam Leonard","family":"Wong","sequence":"additional","affiliation":[{"name":"Department of Statistical Sciences, University of Toronto, Toronto, ON M5S 1A1, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,27]]},"reference":[{"key":"ref_1","unstructured":"Amari, S.I., and Nagaoka, H. 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