2021/10/06 by Noémie C. Combe, Combe, Noemie C.
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Data Visualization and Analytics #Differential Geometry (math.DG) #FOS: Mathematics #Rough Sets and Fuzzy Logic #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2110.02607
openalex publication_date 2021/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
New relations between algebraic geometry, information theory and Topological\nField Theory are developed. One considers models of databases subject to noise\ni.e. probability distributions on finite sets, related to exponential families.\nWe prove explicitly that these manifolds have the structure of a pre-Frobenius\nmanifold, being a pre-structure appearing in the process of axiomatisation of\nTopological Field Theory. On one hand, this allows us to develop relations to\nalgebraic geometry, by proving explicitly that a statistical pre-Frobenius\nmanifold forms an algebraic variety over \ℚ (i.e. \ℚ-toric\nvariety). On the other hand, this allows further developments of recent results\nconcerning the hidden symmetries of those objects. Using classical web theory,\nit has been shown that those symmetries have the structure of Commutative\nMoufang Loops. Our result allows to develop more algebraically this statement,\nin a two-fold way. First, from an algebraic point of view it follows that\nstatistical pre-Frobenius manifolds are equipped with algebraizable webs.\nSecondly, from the differential geometry point of view, it follows that these\nwebs are hexagonal and isoclinic. This statement is important since it directly\nimpacts the geometric properties of the it statistical data, which are\ntightly related to the webs. Hence, this allows deeper connections to the\nbranch of algebraic statistics, which is concerned with the development of\ntechniques in algebraic geometry, commutative algebra, to address problems in\nstatistics and its applications. Examples are provided and discussed.\n