Fisher Information Geometry of the Barycenter of Probability Measures

28/08/2013
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Fisher Information Geometry of the Barycenter of Probability Measures

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Fisher Information Geometry of the Barycenter of Probability Measures Mitsuhiro Itoh and Hiroyasu Satoh Institute of Mathematics, University of Tsukuba, Japan and Tokyo Denki University, Japan Motivation. Consider the following character- ization problem. Let (Xo, go) be a Damek-Ricci space. Let (X, g) be an Hadamard manifold, a simply connected complete Riemannian manifold of nonpositive curvature. Assume (X, g) ∼= (Xo, go) (quasi-isometric). Then, is (X, g) itself Damek- Ricci ? Here (Xo, go) is Damek-Ricci, an R-extention of a generalized Heisenberg group N. A Damek- Ricci space is a solvable Lie group with a left invariant metric. A Damek-Ricci space is Riemannian homogeneous and of nonpos- itive curvature. Moreover, a Damek-Ricci space is harmonic, namely, mean curvature of a geodesic sphere is a function of radius. A Damek-Ricci space is a rank one symmetric space of noncompact type, when it is strictly negative curvature. RHn, CHn, HHn and a Cayley hyperbolic space QH2 exhaust the rank one symmetric spaces of noncompact type. §1 Barycenter and barycenter-isometric maps Denote by ∂X the ideal boundary of (X, g). Let P+(∂X) = P+(∂X, dθ) be the space of probability measures on ∂X, absolutely con- tinuous with respect to the canonical mea- sure dθ and having positive density function. So any µ ∈ P+(∂X) is written as µ(θ) = f(θ)dθ, θ ∈ ∂X, f(θ) > 0. Let Bθ(x) = B(x, θ), x ∈ X, θ ∈ ∂X be the Busemann function on X associated to θ, normalized at a reference point o, defined by Bθ(x) = lim t→∞ {d(x, γ(t)) − t}, where γ(t) denotes the geodesic starting o and going to θ. It holds |∇Bθ(·)| = 1 at any point x. Further we have the so-called Busemann cocycle formula with respect to a Riemannian isom- etry φ of (X, g) Bθ(φx) = Bφ−1θ(x) + Bθ(o) ∀ (x, θ) ∈ X × ∂X See [G-J-T]. Definition 1.1. Let µ ∈ P+(∂X). Then a point y ∈ X is called a barycenter of µ, if the function Bµ : X → R, defined by Bµ(x) = ∫ ∂X Bθ(x)dµ(θ) (1) takes a least value at y. Note the Busemann function and hence the function Bµ(x) is convex in a non-positively curved manifold X. To discuss the existence and uniqueness of barycenter we need fur- ther the strict convexity hypothesis on the Busemann function. Proposition 1.1. Let (X, g) be an Hadamard manifold. Assume that the Hessian DdB(x,θ) of any Busemann function B(x, θ) is strictly positive, except for the gradient direction ∇B(·, θ). Then, there exists uniquely a barycenter for every µ ∈ P+(∂X). See [B-C-G-1, Appendice A] for the proof. We have thus a map bar : P+(∂X, dθ) → X; µ → y, and call it the barycenter map and write y = bar(µ). Note. bar(ˆφ♯µ) = φ(bar(µ)) for a Riemannian isometry φ of (X, g). Here ˆφ denotes the bijective map (homeomorphism) : ∂X → ∂X induced from φ. Remark. in [B-C-G-1] Besson, Courtois and Gallot utilize the notion of barycenter to as- sert the Mostow rigidity of hyperbolic man- ifolds. In fact, let f : ∂X → ∂X be a cer- tain map, where X = RHn, n ≥ 3, a real hyperbolic space. Then, there exists a map F : X → X; F(y) = bar(f∗µy), y ∈ X asso- ciated to the map f, where µy ∈ P+(∂X, dθ) is a special probability measure, called Pois- son kernel probability measure, appeared in §2 and they showed F : X → X is an isometry by using Schwarz’s iequality lemma([B-C-G- 1]). Now, let Φ : ∂X → ∂X be a bijective map (homeomorphism) and Φ♯ : P+(∂X, dθ) → P+(∂X, dθ) be the push-forward map induced from Φ. Φ♯ satisfies from definition of push-forward map ∫ θ∈∂X f(θ) d[Φ♯µ](θ) = ∫ θ∈∂X (f ◦ Φ)(θ) dµ(θ) for any function f = f(θ) on ∂X. Definition 1.2. We consider the following sit- uation: The map Φ♯ yields a bijective map φ : X → X satisfying bar ◦ Φ♯ = φ ◦ bar in the diagram P+(∂X, dθ) Φ♯ −→ P+(∂X, dθ) (2) ↓ bar ↓ bar X φ −→ X We call such a φ a barycenter-isometric map of (X, g) and denote it bar(Φ). Lemma 1.1. The composition φ◦φ1 of barycenter- isometric maps φ = bar(Φ), φ1 = bar(Φ1) is also barycenter-isometric, with φ ◦ φ1 = bar(Φ ◦ Φ1). Proof. With respect to their composition one can check bar(Φ ◦ Φ1) = bar(Φ) ◦ bar(Φ1) (3) Theorem 1.1. Let φ : X → X be a barycenter- isometric map induced from a homeomorphic map Φ : ∂X → ∂X. Assume that φ is of C1, then φ is a Riemannian isometric map of (X, g), i.e., φ fulfills φ∗g = g. (4) For its proof we need the notion of Fisher in- formation geometry together with the Pois- son kernel. §2. Poisson kernel and Fisher Information Geometry Now we assume that an Hadamard manifold (X, g) admits Poisson kernel. Definition 2.1. A function P(x, θ) of (x, θ) ∈ X ×∂X is called Poisson kernel, when (i) it induces the fundamental solution of the Dirichlet problem at the ideal boundary ∆u = 0 on X and u|∂X = f for a given data f ∈ C(∂X) so the solution u is described as u = u(x) = ∫ ∂X P(x, θ)f(θ)dθ, (ii) P(x, θ) > 0 for any (x, θ). Then, the mea- sure P(x, θ)dθ is a probability measure on ∂X parametrized by a point x of X and (iii) P(o, θ) = 1 for any θ(normalization at the ref. point o). (iv) limx→θ1 P(x, θ) = 0, ∀θ, θ1 ∈ ∂X, θ1 ̸= θ A Damek-Ricci space admits a Poisson kernel described specifically as P(x, θ) = exp{−QB(x, θ)}. in terms of B(x, θ) and the volume entropy Q > 0. See [B-C-G-1], [I-S-1], [I-S-2],[I-S-3], [A-B] Lemma 2.1. µx := P(x, θ)dθ ∈ P+(∂X) is a probability measure, parametrized in x for which bar(µx) = x. For a point x ∈ X, let bar−1(x) := {µ ∈ P+(∂X) | bar(µ) = x}. Then, the set bar−1(x) ⊂ P+(∂X) is path- connected and we can discuss the tangent space Tµbar−1(x) to bar−1(x), and then ν ∈ TµP+(∂X) belongs to Tµbar−1(x) if and only if ∫ θ dB(x,θ)(U) dν(θ) = 0 for any tangent vector U ∈ TxX. Now we take µx = P(x, θ)dθ. Then µx ∈ bar−1(x), seen as before. Let Θ : X → P+(∂X); x → µx be the canon- ical map, which we call Poisson kernel map. Proposition 2.1. Let x be a fixed point and U a tangent vector at x. For any ν ∈ Tµxbar−1(x) G(dΘx(U), ν) = 0, where G is the Fisher information metric de- fined on the space P+(∂X). From the proposition we have the fibration of P+(∂X) over the Hadamard manifold X whose fibre over x is bar−1(x). Further the Poisson kernel map Θ : X → P+(∂X) gives a cross section of the fibration. Proof of Proposition 2.1. Since P(x, θ) = exp{−QB(x, θ)}, dΘx(U) = −Q dB(x,θ)(U) µx which we denote by νo. Then, from definition of the Fisher information metric we have Gµx(νo, ν) = ∫ ∂X dνo dµx dν dµx dµx = ∫ −QdB(x,θ)(U)P(x, θ) P(x, θ) × f(θ) P(x, θ) P(x, θ)dθ = −Q ∫ dB(x,θ)(U)dν(θ) which must be zero, since ν = f(θ)dθ belongs to Tµxbar−1(x). Remark. At µx ∈ P+(∂X) the tangent space TµxP+(∂X) is written in an orthogonal direct sum as TµxP+(∂X) = dΘx(TxX) ⊕ Tµxbar−1(x) (5) with respect to the Fisher information metric G. Remark. (5) is valid also with respect to the L2-inner product < f, f1 >= ∫ ∂X f(θ) f1(θ) dθ. Here, the differential of the Poisson kernel map (dΘ)x : TxX → TµxP+(∂X) is injec- tive. In fact, assume that (dΘ)x(U) = 0 in TµxP+(∂X) for U ∈ TxX. Then, this means dB(x,θ)(U)P(x, θ)dθ = 0. Since P(x, θ) > 0, this implies dB(x,θ)(U) = 0 for any θ. To get a conclusion that U = 0 from this we assume U is not zero and then may assume U is unit. Then, we have a geodesic γ(t) = expx tU and hence a point θo = [γ] so dB(x,θo)(U) = −1. This is a contradiction and thus the map dΘx is injective. Proof of Theorem 1.1. For a x ∈ X let y = ϕx, where ϕ = bar(Φ). From definition of barycenter for any µ ∈ bar−1(x) ∫ dB(x,θ)(U) dµ(θ) = 0, ∀U ∈ TxX. Since Φ♯µ ∈ bar−1(y) for µ ∈ bar−1(x), y is a barycenter of Φ♯µ if and only if ∫ dB(y,θ)(V ) d(Φ♯µ)(θ) = 0, ∀V ∈ TyX. Since θ = Φ−1Φθ, from this we have the fol- lowing ∫ dB(y,Φ−1Φθ)(V ) d(Φ♯µ)(θ) = ∫ (Φ♯dB(y,Φθ)(V )dµ)(θ) = ∫ dB(y,Φθ)(V )dµ(θ) = 0 which is valid for any µ ∈ bar−1(x) and in- dicates that dB(y,Φθ)(V )dµ(θ) is orthogonal to the tangent space Tµbar−1(x). In par- ticular, dB(y,Φθ)(V )dµx(θ) belongs from (5) to dΘx(TxX). So, we conclude that for any V ∈ TϕxX there exists U ∈ TxX such that dB(ϕx,Φθ)(V ) = dB(x,θ)(U). The vector V depends on a vector U so we may write V = dϕxU, where dϕx is the defferential map : TxX → TϕxX of the map ϕ. Then, we may assume ⟨∇B(ϕx,Φθ), dϕx(U)⟩ϕx = ⟨∇B(x,θ)), U⟩x, which is reduced into, by using the formal adjoint dϕ∗ x : TϕxX → TxX ⟨dϕ∗ x∇B(ϕx,Φθ), U⟩x = ⟨∇B(x,θ)), U⟩x, for any U. As a consequence of this, the gradient vector fields must satisfy dϕ∗ x∇B(ϕx,Φθ) = ∇B(x,θ) (6) for any x in X and θ ∈ ∂X. Now take an arbitrary unit vector V ∈ TϕxX. So, we have V = ∇B(ϕx,Φθ) for some θ. Then from the above equation we have |dϕ∗ xV | = |dϕ∗ x∇B(ϕx,Φθ)| = |∇B(x,θ)| = 1 where we used |∇B(x,θ)| = 1. This holds for any unit vector so dϕ∗ x and hence dϕx : TxX → TϕxX is a linear isometry and hence ϕ : X → X is a Riemannian isometry of (X, g). §3 Quasi-isometries and quasi-geodesics Let X be an Hadamard manifold with the ideal boundary ∂X. Definition 3.1 Let φ : X → X be a (smooth) map. It is called rough-isometric , or quasi- isometric, when φ satisfies the following, that is, there exist λ > 1 and k > 0 such that for any points x, x′ in X 1 λ d(x, x′) − k < d(φ(x), φ(x′)) < λd(x, y) + k.(7) Note a rough-isometric map is not necessarily continuous. See [Bourd], More generally, a map f : X1 → X2 is called a (λ, k)-quasi-isometric map, if there exist con- stants λ > 1 and k > 0 such that λ−1d1(x, x′) − k < d2(Fx, Fx′) < λd1(x, x′) + k A quasi-isometric map is a generalization of an isometric map. Note we say a (λ, k)-quasi-isometric map sim- ply a quasi-isometric map by abbreviating, when we do not mention the constants λ, k, precisely We say that metric spaces X1 and X2 are quasi-isometric, X1 ∼= X2 (quasi-isometric), if they satisfy one of the following two con- ditions; (i) There exist quasi-isometric maps f : X1 → X2 and g : X2 → X1 and a positive num- ber ε such that g ◦ f and f ◦ g are in an ε- neighborhood of the identity maps idX1 , idX2 , respectively. (ii) There exist a quasi-isometry f : X1 → X2 and ε > 0 such that f(X1) is ε-dense in X2. Let (X, g) be a Riemannian manifold which is quasi-isometric to another Riemannian mani- fold (Xo, go). Then any Riemannian isometry of (Xo, go) induces a bijective quasi-isometric map of (X, g). A curve c : R → X is called a quasi-geodesic, if c is a quasi-isometric map, that is, λ−1|t′ − t| − k < d(c(t), c(t′)) < λ|t′ − t| + k, t, t′ ∈ R for some λ > 1, k > 0. We also call a curve c : [a, b] → X a quasi-geodesic segment, when it satisfies the above inequality in any t, t′ ∈ [a, b]. A geodesic is quasi-geodesic. A quasi-isometric map f : (Xo, go) → (X, g) maps a geodesic γ : R → Xo into a quasi- geodesic f ◦ γ : R → X. Moreover, it holds that let φ : X → X be a quasi-isometric and γ : R → X be a quasi-geodesic. Then the curve φ ◦ γ : R → X is quasi-geodesic. Let F : ∂Xgeod → ∂Xq−geod : [γ]geod → [γ]q−geod(8) be the inclusion map. If an Hadamard mani- fold (X, g) satisfies a certain negative curva- ture condition or a hyperbolicity condition, then the F is bijective. In fact, if the cur- vature satisfies K < −k2 < 0, so is F. See [K-1] for a strictly negative curvature case and [Bourd], [K-2] for the case of manifolds satisfying the hyperbolicity condition. Now we consider the following situation: an Hadamard manifold (X, g) is quasi-isometric with another Hadamard manifold (Xo, go) which is equipped with isometries. An isometry φ of (Xo, go) gives rise of a quasi- isometric bijective map of (X, g). So, φ in- duces a bijective map ˆφ : ∂Xq−geod → ∂Xq−geod, since, for any quasi-geodesic σ, φ◦σ is quasi- geodesic, and if σ ∼ σ1, then φ ◦ σ ∼ φ ◦ σ1. However, ∂Xq−geod is identified with ∂Xgeod = ∂X by the natural map F. So, φ induces a bijective map ˜φ = F ◦ ˆφ ◦ F−1 : ∂X → ∂X. References [A-N] S.Amari and H.Nagaoka, Methods of Information Geometry, AMS,2000. 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