On affine immersions of the probability simplex and their conformal flattening

07/11/2017
Auteurs : Atsumi Ohara
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22629
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Embedding or representing functions play important roles in order to produce various information geometric structure. This paper investigates them from a viewpoint of affine differential geometry [2]. By restricting affine immersions to a certain class, the probability simplex is realized to be 1-conformally flat [3] statistical manifolds immersed in Rn+1. Using this fact, we introduce a concept of conformal flattening of such manifolds to obtain dually flat statistical (Hessian) ones with conformal divergences, and show explicit forms of potential functions, dual coordinates. Finally, we demonstrate applications of the conformal flattening to nonextensive statistical physics and certain replicator equations on the probability simplex.

On affine immersions of the probability simplex and their conformal flattening

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On affine immersions of the probability simplex and their conformal flattening

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Embedding or representing functions play important roles in order to produce various information geometric structure. This paper investigates them from a viewpoint of affine differential geometry [2]. By restricting affine immersions to a certain class, the probability simplex is realized to be 1-conformally flat [3] statistical manifolds immersed in Rn+1. Using this fact, we introduce a concept of conformal flattening of such manifolds to obtain dually flat statistical (Hessian) ones with conformal divergences, and show explicit forms of potential functions, dual coordinates. Finally, we demonstrate applications of the conformal flattening to nonextensive statistical physics and certain replicator equations on the probability simplex.

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On affine immersions of the probability simplex and their conformal flattening Atsumi Ohara University of Fukui, Fukui 910-8507, Japan, ohara@fuee.u-fukui.ac.jp Abstract. Embedding or representing functions play important roles in order to produce various information geometric structure. This paper investigates them from a viewpoint of affine differential geometry [2]. By restricting affine immersions to a certain class, the probability simplex is realized to be 1-conformally flat [3] statistical manifolds immersed in Rn+1 . Using this fact, we introduce a concept of conformal flattening of such manifolds to obtain dually flat statistical (Hessian) ones with con- formal divergences, and show explicit forms of potential functions, dual coordinates. Finally, we demonstrate applications of the conformal flat- tening to nonextensive statistical physics and certain replicator equations on the probability simplex. Keywords: Conformal flattening, affine differential geometry 1 Introduction In the theory of information geometry for statistical models, the logarithmic function is crucially significant to give a standard information geometric struc- ture for exponential family [1]. By changing the logarithmic function to the other ones we can deform the standard structure to new one keeping its basic prop- erty as a statistical manifold, which consists of a pair of mutually dual affine connections (∇, ∇∗ ) with respect to Riemannian metric g. There exists several ways [6, 4, 5] to introduce such freedom of functions to deform statistical mani- fold structure and the functions are sometimes called embedding or representing functions. In this paper we elucidate common geometrical properties of statistical man- ifolds defined by representing functions, using concepts from affine differential geometry [2, 3]. 2 Affine immersion of the probability simplex Let Sn be the probability simplex defined by Sn := { p = (pi) pi ∈ R+, n+1 ∑ i=1 pi = 1 } , where R+ denotes the set of positive numbers. Consider an affine immersion [2] (f, ξ) of the simplex Sn . Let D be the canonical flat affine connection on Rn+1 . Further, let f be an immersion of Sn into Rn+1 and ξ be a transversal vector field on Sn . For a given affine immersion (f, ξ) of Sn , the induced torsion-free connection ∇ and the affine fundamental form h are defined from the Gauss formula by DXf∗(Y ) = f∗(∇XY ) + h(X, Y )ξ, X, Y ∈ X(Sn ), (1) where X(Sn ) is the set of vector fields on Sn . It is well known [3, 2] that the realized geometric structure (Sn , ∇, h) is a statistical manifold if and only if (f, ξ) is non-degenerate and equiaffine, i.e., h is non-degenerate and ∇h is symmetric. Further, a statistical manifold (Sn , ∇, h) is 1-conformally flat [3] (, but not necessarily dually flat nor of constant curvature). Now we consider the affine immersion with the following assumptions. Assumptions: 1. The affine immersion (f, ξ) is nondegenerate and equiaffine, 2. The immersion f is given by the component-by-component and common representing function L, i.e., f : Sn ∋ p = (pi) 7→ x = (xi ) ∈ Rn+1 , xi = L(pi), i = 1, · · · , n + 1, 3. The representing function L : R+ → R is concave with L′′ < 0 and strictly increasing, i.e., L′ > 0. Hence, the inverse of L denoted by E exists, i.e., E ◦ L = id. 4. Each component of ξ satisfies ξi < 0, i = 1, · · · , n + 1 on Sn . Remark 1. From the third assumption, it follows that L′ E′ = 1, E′ > 0 and E′′ > 0. Note that L is concave with L′′ < 0 or convex L′′ > 0 if and only if there exists ξ for h to be positive definite. Hence, we can regard h as a Riemannian metric on Sn . The details are described later. 2.1 Conormal vector and the geometric divergence Define a function Ψ on Rn+1 by Ψ(x) := n+1 ∑ i=1 E(xi ), then f(Sn ) immersed in Rn+1 is expressed as a level surface of Ψ(x) = 1. Denote by Rn+1 the dual space of Rn+1 and by ⟨ν, x⟩ the pairing of x ∈ Rn+1 and ν ∈ Rn+1. The conormal vector [2] ν : Sn → Rn+1 for the affine immersion (f, ξ) is defined by ⟨ν(p), f∗(X)⟩ = 0, ∀X ∈ TpSn , ⟨ν(p), ξ(p)⟩ = 1 (2) for p ∈ Sn . Using the assumptions and noting the relations: ∂Ψ ∂xi = E′ (xi ) = 1 L′(pi) > 0, i = 1, · · · , n + 1, we have νi(p) := 1 Λ ∂Ψ ∂xi = 1 Λ(p) E′ (xi ) = 1 Λ(p) 1 L′(pi) , i = 1, · · · , n + 1, (3) where Λ is a normalizing factor defined by Λ(p) := n+1 ∑ i=1 ∂Ψ ∂xi ξi = n+1 ∑ i=1 1 L′(pi) ξi (p). (4) Then we can confirm (2) using the relation ∑n+1 i=1 Xi = 0 for X = (Xi ) ∈ X(Sn ). Note that v : Sn → Rn+1 defined by vi(p) = Λ(p)νi(p) = 1 L′(pi) , i = 1, · · · , n + 1, also satisfies ⟨v(p), f∗(X)⟩ = 0, ∀X ∈ TpSn . (5) Further, it follows, from (3), (4) and the assumption 4, that Λ(p) < 0, νi(p) < 0, i = 1, · · · , n + 1, for all p ∈ Sn . It is known [2] that the affine fundamental form h can be represented by h(X, Y ) = −⟨ν∗(X), f∗(Y )⟩, X, Y ∈ TpSn . In our case, it is calculated via (5) as h(X, Y ) = −Λ−1 ⟨v∗(X), f∗(Y )⟩ − (XΛ−1 )⟨v, f∗(Y )⟩ = − 1 Λ n+1 ∑ i=1 ( 1 L′(pi) )′ L′ (pi)Xi Y i = 1 Λ n+1 ∑ i=1 L′′ (pi) L′(pi) Xi Y i . Since h is positive definite from the assumptions 3 and 4, we can regard it as a Riemannian metric. Utilizing these notions from affine differential geometry, we can introduce the function ρ on Sn × Sn , which is called a geometric divergence [3], as follows: ρ(p, r) = ⟨ν(r), f(p) − f(r)⟩ = n+1 ∑ i=1 νi(r)(L(pi) − L(ri)) = 1 Λ(r) n+1 ∑ i=1 L(pi) − L(ri) L′(ri) , p, r ∈ Sn . (6) We can easily see that ρ is a contrast function [7, 1] of the geometric structure (Sn , ∇, h) because it holds that ρ[X|] = 0, h(X, Y ) = −ρ[X|Y ], (7) h(∇XY, Z) = −ρ[XY |Z], h(Y, ∇∗ XZ) = −ρ[Y |XZ], (8) where ρ[X1 · · · Xk|Y1 · · · Yl] stands for ρ[X1 · · · Xk|Y1 · · · Yl](p) := (X1)p · · · (Xk)p(Y1)r · · · (Yl)rρ(p, r)|p=r for p, r ∈ Sn and Xi, Yj ∈ X(Sn ). 2.2 Conformal divergence and conformal transformation Let σ be a positive function on Sn . Associated with the geometric divergence ρ, the conformal divergence [3] of ρ with respect to a conformal factor σ(r) is defined by ρ̃(p, r) = σ(r)ρ(p, r), p, r ∈ Sn . The divergence ρ̃ can be proved to be a contrast function for (Sn , ˜ ∇, h̃), which is conformally transformed geometric structure from (Sn , ∇, h), where h̃ and ˜ ∇ are given by h̃ = σh, (9) h( ˜ ∇XY, Z) = h(∇XY, Z) − d(ln σ)(Z)h(X, Y ). (10) When there exists such a positive function σ that relates (Sn , ∇, h) with (Sn , ˜ ∇, h̃) as in (9) and (10), they are said 1-conformally equivalent and (Sn , ˜ ∇, h̃) is also a statistical manifold [3]. 2.3 A main result Generally, the induced structure (Sn , ˜ ∇, h̃) from the conformal divergence ρ̃ is not also dually flat, which is the most abundant structure in information geom- etry. However, by choosing the conformal factor σ carefully, we can demonstrate (Sn , ˜ ∇, h̃) is dually flat. Hereafter, we call such a transformation as conformal flattening. Define Z(p) := n+1 ∑ i=1 νi(p) = 1 Λ(p) n+1 ∑ i=1 1 L′(pi) , then it is negative because each νi(p) is. The conformal divergence to ρ with respect to the conformal factor σ(r) := −1/Z(r) is ρ̃(p, r) = − 1 Z(r) ρ(p, r). Proposition 1. If the conformal factor is given by σ = −1/Z, then statistical manifold (Sn , ˜ ∇, h̃) that is 1-conformally transformed from (Sn , ∇, h) is dully flat and ρ̃ is canonical where mutually dual potential functions and coordinate systems are explicitly given by θi (p) = xi (p) − xn+1 (p) = L(pi) − L(pn+1), i = 1, · · · , n (11) ηi(p) = Pi(p) := νi(p) Z(p) , i = 1, · · · , n, (12) ψ(p) = −xn+1(p) = −L(pn+1), (13) φ(p) = 1 Z(p) n+1 ∑ i=1 νi(p)xi (p) = n+1 ∑ i=1 Pi(p)L(pi). (14) Proof) Using given relations, we first show that the conformal divergence ρ̃ is the canonical divergence for (Sn , ˜ ∇, h̃): ρ̃(p, r) = − 1 Z(r) ⟨ν(r), f(p) − f(r)⟩ = ⟨P(r), f(r) − f(p)⟩ = n+1 ∑ i=1 Pi(r)(xi (r) − xi (p)) = n+1 ∑ i=1 Pi(r)xi (r) − n ∑ i=1 Pi(r)(xi (p) − xn+1 (p)) − (n+1 ∑ i=1 Pi(r) ) xn+1 (p) = φ(r) − n ∑ i=1 ηi(r)θi (p) + ψ(p). (15) Next, let us confirm that ∂ψ/∂θi = ηi. Since θi (p) = L(pi) + ψ(p), i = 1, · · · , n, we have pi = E(θi − ψ), i = 1, · · · , n + 1, by setting θn+1 := 0. Hence, we have 1 = n+1 ∑ i=1 E(θi − ψ). Differentiating by θj , we have 0 = ∂ ∂θj n+1 ∑ i=1 E(θi − ψ) = n+1 ∑ i=1 E′ (θi − ψ) ( δi j − ∂ψ ∂θj ) = E′ (xj ) − (n+1 ∑ i=1 E′ (xi ) ) ∂ψ ∂θj . This implies that ∂ψ ∂θj = E′ (xj ) ∑n+1 i=1 E′(xi)