Fisher Information Geometry of the Barycenter Map

21/09/2014
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Fisher Information Geometry of the Barycenter Map

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Fisher Information Geometry of The Barycenter Map Mitsuhiro Itoh (Institute of Mathematics, University of Tsukuba, Japan)∗1 Hiroyasu Satoh (Nippon Institute of Technology, Japan)∗2 §1. Introduction We would like to report Fisher information geometry of the barycenter map asso- ciated with normalized Busemann function Bθ of an Hadamard manifold X, simply connected non-positively curved manifold and to present an application to Rieman- nian geometry of X from viewpoint of Fisher information geometry. This report is an improvement of [ItSat’13] together with a fine investigation of the barycenter map. §2. The barycenter map Let µ be a probability measure on the ideal boundary ∂X of X. A point x ∈ X is called a barycenter of µ, when x is a critical point of the µ-average Busemann function on X; Bµ(y) = ∫ θ∈∂X Bθ(y)dµ(θ), y ∈ X. Denote by P+ = P+ (∂X, dθ) the space of probability measures µ = f(θ)dθ defined on ∂X satisfying µ ≪ dθ and with continuous density f = f(θ) > 0. A point x ∈ X is a barycenter of a measure µ if and only if the µ-average one form dBµ(·) =∫ θ∈∂X dBθ(·)dµ(θ) vanishes at x. We follow the idea given by [DoEa], [BeCoGa’95]. Theorem 2.1([ItSat’14-2]). The function Bµ admits for any µ ∈ P+ a barycenter, provided (i) X satisfies the axiom of visibility and (ii) Bθ(x) is continuous in θ ∈ ∂X. X is said to satisfy the axiom of visibility, when any two ideal points θ, θ1 of ∂X, θ ̸= θ1, can be joined by a geodesic in X (see [EO]). In [BeCoGa’95] the existence theorem is verified under the condtions that (i) Bθ satisfies limx→θ1 Bθ(x) = +∞, when θ1 ̸= θ and (ii) Bθ(·) is continuous with respect to θ. The condition (i) can be replaced by the axiom of visibility (refer to [BGS]) to obtain Theorem 2.1. For the uniqueness we have Theorem 2.2([ItSat’14-2]). Assume (i) and (ii) in Theorem 2.1. If, for some µo ∈ P+ the µo-average Hessian (∇d Bµo )x(·, ·) = ∫ θ∈∂X (∇d Bθ)x(·, ·)dµo(θ) is positive definite on TxX at any x ∈ X, then the existence of barycenter is unique for any µ ∈ P+ . So, we obtain a map, called the barycenter map bar : P+ = P+ (∂X, dθ) → X, µ → x, where x is a barycenter of µ. Notice that the differentiability of Bµ is guaranteed when the Hessian of Bθ is uniformly bounded with respect to θ. ∗1 e-mail: itohm@math.tsukuba.ac.jp ∗2 e-mail: hiroyasu@nit.ac.jp §3. A fibre space structure of P+ over X and Fisher information metric It is easily shown that the map bar is regular at any µ, that is, the differential map d barµ : TµP+ → TyX is surjective(see [BeCoGa’96]). Moreover the map bar is itself surjective and hence it yields a fibre space projection with fibre bar−1 (x) over x ∈ X, P+ (∂X, dθ) (1) ↓ bar X provided X carries Busemann-Poisson kernel P(x, θ)dθ = exp{−qBθ(x)}, the funda- mental solution of Dirichlet problem at the boundary ∂X, namely, Poisson kernel represented by Bθ(x) in an exponential form (q = q(X) > 0 is the volume entropy of X). An Hadamard manifold admitting Busemann-Poisson kernel turns out to be asymptotically harmonic ([Led],[ItSat’11]), since ∆Bθ is constant for any θ. The tangent space Tµbar−1 (x) of bar−1 (x) is characterized as {τ ∈ TµP+ | ∫ θ∈∂X (dBθ)x(U)dτ(θ) = 0, ∀U ∈ TxX} so one gets Proposition 3.1. τ ∈ TµP+ belongs to Tµbar−1 (x) if and only if Gµ (τ, Nµ(U)) = 0, ∀U ∈ TxX (2) where Gµ is the Fisher information metric of P+ at µ and Nµ : TxX → TµP+ is a linear map defined by Nµ : TxX → TµP+ U → (dBθ)x(U)dµ(θ). From this we have Proposition 3.2. At any µ ∈ P+ the tangent space TµP+ admits an orthogonal direct sum decomposition into the vertical and horizontal subspaces as TµP+ = Tµbar−1 (x) ⊕ ImNµ, x = bar(µ), with dim ImNµ = dim X. Definition 3.1([AN],[Fr] and [ItSat’11]). A positive definite inner product Gµ on the tangent space TµP+ is defined by Gµ(τ, τ1) = ∫ θ∈∂X dτ dµ (θ) dτ1 dµ (θ)dµ(θ), τ, τ1 ∈ TµP+ . The collection {Gµ | µ ∈ P+ } provides a Riemannian metric on P+ , called Fisher information metric G. As the G is viewed as a Riemannian metric on an infinite dimensional manifold P+ , the Levi-Civita connection ∇ is given (see p.276, [Fr]) ∇τ1 τ = − 1 2 ( dτ dµ (θ) dτ1 dµ (θ) − ∫ dτ dµ (θ) dτ1 dµ (θ)dµ(θ) ) µ, (3) at a point µ ∈ P+ for constant vector fields τ, τ1 on P+ . The space P+ with the metric G has then constant sectional curvature 1 4 (refer to Satz 2, §1, [Fr]). By using the formula (3) we have Theorem 3.3. Let γ(t) be a geodesic in P+ of γ(0) = µ and γ′ (0) = τ ∈ TµP+ , a unit tangent vector. Then γ(t) is described as γ(t) = { cos2 ( t 2 ) + 2 cos( t 2 ) sin( t 2 ) dτ dµ (θ) + sin2 ( t 2 ) ( dτ dµ )2 (θ) } dµ(θ) (4) = { cos( t 2 ) + sin( t 2 ) dτ dµ (θ) }2 dµ(θ). Note that the geodesic lies inside of P+ as far as the density is positive with respect to θ ∈ ∂X. Corollary 3.4. Every geodesic in P+ is periodic of period 2π. The length ℓ of a geodesic segment joining two measures µ and µ1 of P+ is given by cos ℓ 2 < ∫ ∂X √ dµ1 dµ (θ)dµ(θ) = ∫ ∂X √ dµ dµ1 (θ)dµ1(θ) (5) and equality “ = ” in (5) holds provided at least cos(ℓ 2 ) + sin(ℓ 2 )dτ dµ (θ) > 0 for any θ. For these see also p. 279, [Fr]. The integration in RHS of (5) is the f-divergence Df (µ||µ1) = ∫ f(dµ1 dµ )dµ, f(u) = √ u in statistical models (refer to p. 56,[AN]). The formula (4), an improvement of the formula given by T. Friedrich (refer to p.279, [Fr]), can then assert Corollary 3.5. Let γ(t) = expµ tτ be a geodesic of γ(0) = µ and γ′ (0) = τ. Then γ is entirely contained in the fibre bar−1 (x) over x = bar(µ) if and only if τ satisfies at µ Gµ(∇τ τ, Nµ(U)) = 0, ∀U ∈ TxX, (6) The condition (6) implies that the tangent vector τ is a totally geodesic vector with respect to the second fundamental form H, i.e., H(τ, τ) = 0 at µ, since the image ImNµ of the linear map Nµ distributes a normal bundle of bar−1 (x) at any measure. Here, Hµ(τ, τ1) := (∇τ τ1)⊥ at µ. Example 3.1. Let o be the base point for ∂X, dim X ≥ 2 such that ∂X ∼= SoX and bar(µ) = o for the canonical measure µ = dθ ∈ P+ . Identify (dBθ)o with − ∑ i θi ei, θi ∈ R, with respect to an orthonormal basis {ei} of ToX. Define τ = 1 c θi θj dθ, i ̸= j a vector tangent to P+ (c is a constant normalizing τ as a unit). Then τ ∈ Tµbar−1 (o) is seen and γ(t) = expµ tτ is a geodesic which is, from Corollary 3.5, contained in bar−1 (o) at least for t, provided the density function is positive. In fact, the τ satisfies (6). §4. Barycentrically associated maps and isometries of X A Riemannian isometry φ of X transforms every geodesic into a geodesic and hence induces naturally a map ˆφ : ∂X → ∂X, a homeomorphism with respect to the cone topology. Further the normalized Busemann function admits a cocycle formula ([GJT]); Bθ(φx) = Bˆφ−1θ(x) + Bθ(φo), ∀(x, θ) ∈ X × ∂X (7) (o is the normalization point of Bθ). Proposition 4.1 (Equivariant action formula). bar ◦ ˆφ♯ = φ ◦ bar, namely (8) bar( ˆφ♯µ) = φ(bar(µ)) ∀µ ∈ P+ , where Φ♯ : P+ → P+ is the push-forward of a homeomorphism Φ of ∂X; ∫ θ∈∂X h(θ) d[Φ♯µ](θ) = ∫ θ∈∂X (h ◦ Φ)(θ) dµ(θ) (9) for any function h = h(θ) on ∂X (see p.4, [V]). So, we consider the situation converse of Proposition 4.1 as Definition 4.1. Let Φ : ∂X → ∂X be a homeomorphism of ∂X. Then, a map φ : X → X is called barycentrically associated to Φ, when φ satisfies the relation bar ◦ Φ♯ = φ ◦ bar in the diagram P+ (∂X, dθ) Φ♯ −→ P+ (∂X, dθ) (10) ↓ bar ↓ bar X φ −→ X So an isometry φ is a map barycentrically associated to Φ = ˆφ Let bar : P+ → X be the barycenter map. Then, with respect to a homeomorphism Φ : ∂X → ∂X and a map φ : X → X we obtain the following ([ItSat’14],[ItSat’14-2]) Theorem 4.2. Assume that a pair (Φ, φ) with φ ∈ C1 satisfies; (a) bar(Φ♯µ) = φ(bar(µ)), ∀µ ∈ P+ , and (b) Θ(φ(x)) = Φ♯ (Θ(x)) , ∀x ∈ X; P+ (∂X, dθ) Φ♯ −→ P+ (∂X, dθ) (11) ↑ Θ ↑ Θ X φ −→ X Then φ must be a Riemannian isometry of X. Here, Θ : X → P+ ; y → P(y, θ)dθ is a map associated with a Busemann-Poisson kernel P(x, θ) = exp{−q Bθ(x)}. Remark 4.1. If X admits a Busemann-Poisson kernel, then Θ gives a cross section of the fibre space P+ → X, since bar(µx) = x for µx = P(x, θ)dθ, and moreover every µ ∈ P+ admits a unique barycenter from Theorem 2.2, since it holds ∫ ∂X (∇dBθ)x(U, V )dµx(θ) = q ∫ ∂X (dBθ)x(U)(dBθ)x(U)dµx(θ), U, V ∈ TxX that is (∇d Bµx )x(U, V ) = q Gµx (Nµx (U), Nµx (V )) (q > 0 is the volume entropy of X) and at any y ∈ X (∇d Bµx )y(U, U) ≥ C (∇d Bµx )x(U, U) for some constant C > 0, depending on x, y. From these the µx-average Hessian ∇d Bµx turns out to be positive definite everywhere. With respect to the conditions (a) and (b) of Theorem 4.2 we have Theorem 4.3. Let X be an Hadamard manifold satisfying assumptions (i) and (ii) of Theorem 2.1 and admit a Busemann-Poisson kernel. Let Φ : ∂X → ∂X be a homeomorphism. If a map φ : X → X is C1 with surjective differential dφx, ∀x ∈ X, then (b) implies (a). §5. Damek-Ricci spaces and motivation A Damek-Ricci space is a solvable Lie group, an R-extension of a generalized Heisen- berg group and carries a left invariant Riemannian metric and further provides a space on which harmonic analysis is developed ([ADY],[DamR]). For precise definition and differential geometry of Damek-Ricci space refer to [BTV]. Damek-Ricci spaces are Hadamard manifolds whose typical examples are rank one symmetric spaces of non- compact type, complex hyperbolic, quaternionic hyperbolic and Cayley hyperbolic spaces as strictly negative curved ones, except for real hyperbolic spaces ([D],[L]). Any Damek-Ricci space satisfies the axiom of visibility and has θ-continous Busemann function (refer to [ItSat’10]) . Moreover, it admits a Busemann-Poisson kernel (see [ItSat’10]) so that it satisfies (i) and (ii) of Theorem 2.1. Most important implication of Damek-Ricci spaces is that they provides counterexample of Lichnerowicz conjecture of non-compact version (refer to [BTV]). So, relating to this, our motivation is to characterize Damek-Ricci spaces from a viewpoint of geometry, since only a Lie group characterization of Damek-Ricci space is known from Heber’s theorem ([Heb]). A Damek-Ricci space turns out recently to be Gromov-hyperbolic, whereas it admits zero sectional curvature (see [ItSat’14-2] for this and refer to [CDP],[Bourd],[K] for the Gromov hyperbolicity). Thus, we pose the following. Let Xo be a Damek-Ricci space and X an Hadamard manifold quasi-isometric to Xo. Assume that X admits a Busemann-Poisson kernel. Then, is X isometric, or homothetic to Xo as a Riemannian manifold ? At least from this assumption we have that any Riemanian isometry of Xo induces a homeomorphism of ∂X of X (for the detail, see [ItSat’14-2]). References [AN] S. Amari, H. Nagaoka, Methods of Information Geometry, Trans. Math. Monogr., 191, AMS, Oxford, 2000. [ADY] J.-Ph. Anker, E. Damek and C. Yacoub, Spherical analysis on harmonic AN groups, Ann. Scuola Norm.Sup.Pisa Cl. Sci.(4), 23 (1996) 643-679. [BTV] J. Berndt, F. Tricerri and L. Vanhecke, Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces, Lecture Notes, 1598, Springer, 1991. [BGS] W. Ballmann,M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Prog. Math., 61, Birkh¨auser, Boston, 1985. [BeCoGa’95] G. Besson, G. Courtois and S. Gallot, Entropes et Rigidit´es des Espaces Localement Sym´etriques de Courbure Strictement N´egative, GAFA, 5 (1995) 731-799. [BeCoGa’96] G. Besson, G. Courtois and S. Gallot, A simple and constructive proof of Mostow’s rigidity and the minimal entropy theorems, Erg. Th. Dyn. Sys., 16 (1996), 623-649. [Bourd] M. Bourdon, Structure conforme au bord et flot g´eod´esique d’un CAT(-1)- espace, L’Enseignement Math., 41 (1995), 63-102. [CDP] M. Coornaert, T. Delzant and A. Papadopoulos, G´eom´etrie et th´eorie des groupes, Lecture Notes, 1441, Springer-Verlag, 1990. [Dam-R] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal., 2 (1992), 213-248. [D], I. Dotti, On the curvature of certain extensions of H-type groups, Proc. A.M.S., 125 (1997), 573-578. [DoEa] E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math., 157 (1986), 23-48. [EO] P. Eberlein, B. O’Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-110. [Fr] T. Friedrich, Die Fisher Information und symplektische Structuren, Math. Nachrich., 153 (1991), 273-296. [GJT] Y. Guivarc’h, L. Ji and J.C. Taylor, Compactifications of Symmetric Spaces, Birkh¨auser, 1997. [Heb] J. Heber, On harmonic and asymptotically harmonic homogeneous spaces, Geom. Funct. Anal., 16 (2006), 869-890. [ItSat’10] M. Itoh and H. Satoh, Information geometry of Poisson kernel on Damek- Ricci spaces, Tokyo J. 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