#
Geometry of Fisher Information Metric and the Barycenter Map^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**R**

^{n}, and hence to an open ball D

^{n}, whose actual boundary is S

^{n−}

^{1}. X admits also the ideal boundary ∂X as a quotient space of oriented geodesics on X. Then, we are able to consider Dirichlet problem at boundary ∂X; given a f ∈ C

^{0}(∂X), find a solution u = u(x) on X satisfying Δu = 0, u|

_{∂X}= f. Using the fundamental solution P = P (x, θ), called Poisson kernel, when its existence is guaranteed, the solution is described as

**Theorem 1**([4–6]). Let (X, g) be an n-dimensional Damek-Ricci space. Then the map Θ is homothetic with respect to the Fisher metric G and g;$\mathrm{\Theta}*G=\frac{Q}{n}g$ where Q is volume entropy of (X, g). Further Θ is a harmonic map.

**Theorem 2**([5]). Let (X, g) be an Hadamard manifold which is equipped with Poisson kernel P (x, θ). Assume that the map$\mathrm{\Theta}:X\to \mathcal{P}(\partial X)$ is homothetic; Θ

^{*}G = C g, C > 0, and harmonic with respect to the metrics G and g. Then (X, g) is asymptotically harmonic and satisfies visibility axiom. Moreover, C = Q/n and the Poisson kernel has the form P (x, θ) = exp{−Q B

_{θ}(x)} in terms of Busemann function B

_{θ}on X.

## 2. Main Results and Conclusive Remarks

_{μ}tτ on $\mathcal{P}(M)$ in a simple form (Theorem 9);

**R**. Moreover, from this formula which is an improvement of the formula given by T. Friedrich ([2]) we obtain

**Theorem 3.**Let μ and μ

^{*}be arbitrary distinct probability measures in$\mathcal{P}(M)$. Then, a curve$t\in \mathbf{R}\mapsto \mu (t)\in \mathcal{P}(M)$defined by

^{*}, μ ≠ μ

^{*}can be joined by a unique geodesic. The quantity ℓ = ℓ(μ, μ*), 0< ℓ

_{<}π

_{i}s defined as

**B**

_{μ}: X →

**R**;

_{1}, …, y

_{n}be points of a Euclidean space

**R**

^{3}and μ

_{1}, …, μ

_{n}be non-negative real numbers satisfying ${\sum}_{i}{\mu}_{i}=1$. A point p of

**R**

^{3}is called a barycenter of y

_{i}, i = 1, …, n of weights μ

_{i}, i = 1, …, n, when p satisfies

**R**

^{3}; f :

**R**

^{3}→

**R**; $f(q)={\displaystyle {\sum}_{i=1}^{n}{\mu}_{i}{d}^{2}(q,{y}_{i})}$.

**R**

^{3}with weights with respect to the square-distance can be generalized as one for points of

**R**

^{3}distributed continuously over a bounded set D of

**R**

^{3};

^{2}(x, y) is not essential. Convexity of testing function is crucial in a theory of barycenter. In our study we deal with barycenter with respect to Busemann function, a convex testing function, by following the idea of Douady, Earle ([16]) and Besson, Courtois and Gallot ([8,17]). Refer to [18,19] for studies and results on barycenter of square-distance and of distance over a Riemannian manifold. Refer also to [20] in this direction, which is a reference comment due to Professor M. Gromov at the conference.

_{θ}(x) on X is continuous with respect to θ ∈ ∂X. Uniqueness of barycenter for any μ is assured, when, for some μ

_{0}, average Hessian $\nabla d{\mathrm{B}}_{{\mu}_{0}}$of ${\mathrm{B}}_{{\mu}_{0}}$ is positive definite everywhere on X (Proposition 6). Thus, we have the barycenter map bar : $\mathcal{P}(\partial X)\to X;\mu \mapsto y$, by assigning to μ a barycenter point y of μ. This map turns out to be surjective, when (X, g) admits Busemann-Poisson kernel P (x, θ) = exp{−Q B

_{θ}(x)}, Poisson kernel of Busemann type. Denote by μ

_{x}the probability measure P (x, θ) dθ. Then, bar(μ

_{x}) = x for any x ∈ X. Busemann-Poisson kernel ensures also the uniqueness of barycenter for any μ from the identity (see Theorem 14)

^{−1}(x); x ∈ X}. The image of the linear map ${v}_{x}^{\mu}$ for μ and x = bar(μ) yields a subspace of ${T}_{\mu}\mathcal{P}(\partial X)$, normal to T

_{μ}bar

^{−1}(x), the subspace tangent to a fibre bar

^{−1}(x) so that ${T}_{\mu}\mathcal{P}(\partial X)$ splits into a G-orthogonal direct sum (Theorem 15);

_{μ}bar

^{−1}(x) and ${\text{Imv}}_{x}^{\mu}$ are the vertical, horizontal subspaces of ${T}_{\mu}\mathcal{P}(\partial X)$, respectively. Here, dim ${\text{dimImv}}_{x}^{\mu}=\mathrm{dim}X$. Remark that the fibration asserted here is infinitesimal.

^{−1}(x), x ∈ X is a path-connected submanifold of $\mathcal{P}(\partial X)$. Its geometry is investigated in terms of the second fundamental form;

_{τ}τ

_{1}, the covariant derivative of τ

_{1}in direction to τ with respect to the Levi–Civita connection ∇. Refer to [22,23] for definition of the second fundamental form. Applying the results concerning geodesics on $\mathcal{P}(\partial X)$ given in Section 3 to a submanifold bar

^{−1}(x), we are able to determine a geodesic μ(t) = exp

_{μ}tτ which is entirely contained in bar

^{−1}(x) as

**Theorem 4.**Let μ(t) = exp

_{μ}tτ be a unit speed geodesic, of μ(0) = μ, μ˙(0)=τ, |τ|

_{G,μ}= 1. Then, μ(t) lies entirely on fibre bar

^{−1}(x) if and only if, μ ∈ bar

^{−1}(x), τ ∈ T

_{μ}bar

^{−1}(x) and H

_{μ}(τ, τ) = 0.

_{μ}(τ, τ) = 0 on τ is written down in a manner of information geometry as ${\int}_{\theta \in \partial X}{(d{B}_{\theta})}_{x}(u){(d\tau /d\mu )}^{2}(\theta )d\mu (\theta )=0$.

**Theorem 5.**Let μ, μ* ∈ bar

^{−1}(x). Then, a geodesic joining μ and μ* is contained completely in the same fibre bar

^{−1}(x) if and only if μ and μ* fulfill

_{x}X.

_{μ}satisfies a cocycle formula with respect to ϕ;

^{−1}is the inverse of ϕ, and $\widehat{\varphi}:\partial X\to \partial X$is a ∂X-extension of ϕ and ${\widehat{\varphi}}_{\#}$ is a push-forward induced by $\widehat{\varphi}$. See Theorem 18. From this formula we have

^{−1}(x) is mapped by ${\widehat{\varphi}}_{\#}$ to a fibre bar

^{−1}(ϕx) over ϕx.

_{#}μ) = ϕ(bar(μ)) for any $\mu \in \mathcal{P}(\partial X)$ (we call such a map ϕ as barycentrically associated to Φ).

**Theorem 6.**Let φ : X → X be a C

^{1}-map barycentrically associated to a homeomorphism Φ : ∂X → ∂X. Assume that there exists a cross section$\mathrm{\Sigma}:X\to \mathcal{P}(\partial X)$of the fibre space bar : $\mathcal{P}(\partial X)\to X$, a map satisfying bar ○ Σ = id

_{X}such that a fibrewise diagram commutes

_{θ}(x)} dθ, where P (x, θ) is a Busemann-Poisson kernel on (X, g). The differential map (Θ

_{*})

_{x}of Θ fulfills ${({\mathrm{\Theta}}_{*})}_{x}(u)=-Q\phantom{\rule{0.5em}{0ex}}{v}_{x}^{{\mu}_{x}}(u),\phantom{\rule{0.5em}{0ex}}u\in {T}_{x}X$ in terms of the linear map ${v}_{x}^{{\mu}_{x}}$, μ

_{x}:= P (x, θ)dθ, so that we have

**Corollary 1**([24]). Let Φ : ∂X → ∂X be a homeomorphism of ∂X and φ : X → X be a C

^{1}-map. Assume the following diagram commutes with respect to Poisson kernel map Θ;

## 3. A Space of Probability Measures and Fisher Metric

#### 3.1. A Space of Probability Measures

- P (A) ≥ 0 for any $A\in \mathcal{B}(M)$, P (M) = 1 and P (∅) = 0.
- Let {E
_{j}| j = 1, …, } be a countable sequence of sets of $\mathcal{B}(M)$ satisfying E_{i}∩ E_{j}= ∅ for any i, j, i ≠ j. Then$$P\left({\displaystyle \underset{i=1}{\overset{\infty}{\cup}}{E}_{i}}\right)={\displaystyle \sum _{i=1}^{\infty}P({E}_{i}).}$$

^{−1}= {x ∈

**R**

^{n}; |x| = 1} be a unit (n−1)-sphere and let dω be the standard (n − 1)-spherical volume measure on S

^{n}

^{−1}. Then dθ = (1/A

_{n}

_{−1}) dω is the normalized measure, where A

_{n}

_{−1}is the volume of S

^{n}

^{−1}.

_{1}be probability measures on M. μ is called to be absolutely continuous with respect to μ

_{1}, if μ(A) = 0 for any $A\in \mathcal{B}(M)$, whenever μ

_{1}(A) = 0.

^{1}(M, dθ) such that μ is represented as μ = p dθ, namely μ satisfies

^{0}(M), p(x) > 0 for any x ∈ M.

^{2}-function space L

^{2}(M; dθ) as

_{i}} of $\mathcal{P}(M)$ does not necessarily admit a limit inside $\mathcal{P}(M)$.

_{1}be probability measures in $\mathcal{P}(M)$. Then we can join μ and μ

_{1}by a path μ(t) = (1−t)μ+tμ

_{1}, t ∈ [0, 1] inside $\mathcal{P}(M)$.

_{1}− μ = (p

_{1}− p) dθ satisfying

^{1}in t for any fixed x ∈ M. So, c = c(t) has velocity vector field along c

#### 3.2. Fisher Metric

**Definition 1.**A positive definite inner product G

_{μ}on${T}_{\mu}\mathcal{P}(M)$ at$\mu \in \mathcal{P}(M)$ is defined as

_{G, μ}.

^{x}A dμ(x). See [27]. We represent (10) in an integral form as

**R**.

_{#}μ coincides with (Φ

^{−1})

^{*}μ, the pull-back of μ by the inverse diffeomorphism Φ

^{−1}. Notice that

_{#}: $\mathcal{P}(M)\to \mathcal{P}(M)$ has differential map

_{#}τ is defined similarly as (10). In fact, we have

**Theorem 7**([2]). Let Φ

_{#}be a push-forward. Then it acts on$\mathcal{P}(M)$ isometrically with respect to the Fisher metric G. Namely,

**Proof.**We write μ = p(x) dθ(x) and τ = q(x) dθ(x), τ

_{1}= q

_{1}(x) dθ(x). Set σ = Φ

_{#}μ. We have then from definition of push-forward

_{μ}(τ, τ

_{1}) of τ and τ

_{1}at μ.□

**Note.**The following is known. For any $\mu \in \mathcal{P}(M)$ there exists a homeomorphism Φ : M → M satisfying μ = Φ

_{#}dθ (refer to [28,29]). This fact implies that the action of Homeo(M), which is the group of homeomorphisms on M, on the space $\mathcal{P}(M)$ is isometric and transitive.

**Remark 1.**The embedding ρ given in (8) satisfies$\rho *{\langle \cdot ,\cdot \rangle}_{{L}^{2}}=\frac{1}{4}{G}_{\mu}$, that is,

^{2}-inner product of L

^{2}(M; dθ);

#### 3.3. Levi–Civita Connection

**Theorem 8.**Let τ, τ

_{1}be constant vector fields on P(M). Then, the Levi–Civita connection of the Fisher metric G at μ ∈ P(M) is represented as

**Proof.**Recall that on a Riemannian manifold N with a Riemannian metric g the Levi–Civita connection ∇ of g is an affine connection on N, that is, ∇ is a bilinear map; (Y, Z) ↦ ∇

_{Y}Z satisfying

_{Y}g)(Z, W ) = 0 together with a symmetry condition, that is, the torsion tensor T (Y, Z) := ∇

_{Y}Z − ∇

_{Z}Y −[Y, Z] vanishes. Then, the Levi–Civita connection ∇ exists uniquely and one has Koszul’s formula for ∇

_{1}and τ

_{2}

_{1}] = [τ, τ

_{2}] = [τ

_{1}, τ

_{2}] = 0.

_{2}be a curve in $\mathcal{P}(M)$ of μ(0) = μ and $\dot{\mu}(t)={\tau}_{2}$. We have then

_{μ}G(τ

_{1}, τ

_{2}), (τ

_{1})

_{μ}G(τ, τ

_{2}) and then finally

_{1}is arbitrary, (13) is derived. □

**Theorem 9.**The Riemannian curvature tensor R of the Fisher metric G satisfies

_{1}, τ

_{2}. Hence, sectional curvature of any section τ ∧ τ

_{1}is$K(\tau \wedge {\tau}_{1})=\frac{1}{4}$.

**R**.

#### 3.4. Geodesics

**Theorem 10.**Let μ ∈ P(M) and$\tau \in {T}_{\mu}\mathcal{P}(M)$. Assume τ is a unit tangent vector at μ, i.e., |τ|

_{G,μ}= 1. Then, the geodesic μ(t), denoted by exp

_{μ}tτ, with μ(0) = μ, $\dot{\mu}\left(0\right)=\tau $has the form represented by

**Note**. Set t = π into (15). Then $\mu (\mathrm{\pi})={\left(\frac{d\tau}{d\mu}(x)\right)}^{2}\mu =\frac{q{(x)}^{2}}{p(x)}d\theta $. However, τ is a tangent vector to P(M) so τ satisfies ${\int}_{x\in M}q(x)d\theta (x)=0$ from which there exists a point x

_{o}∈ M with q(x

_{o}) = 0 and then the density function of μ(π) vanishes at point x

_{o}and then $\mu (\mathrm{\pi})\notin \mathcal{P}(M)$.

**Lemma 1.**Let μ(t) = p(x, t) dθ be a geodesic such that μ(0) = μ = p

_{0}(x) dθ, $\dot{\mu}\left(0\right)=\tau =q(x)d\theta \in {T}_{\mu}{\mathcal{P}}^{+}(M)$, |τ|G; μ = 1. Then,

**Proof of Lemma**1. A proof is given in [2]. However, we will give a proof for a later convenience in proving Theorems 16 and 17. So, for simplicity we write by abbreviation μ(t) = p(t) dθ and $\dot{\mu}\left(t\right)=\dot{p}\left(t\right)d\theta $. Letting τ be a constant vector field, we have

_{μ}

_{˙(}

_{t}

_{)}τ with respect to μ(t) is

**Remark 2.**Equation (19) is expressed as

_{0}(x), p˙(x, 0) = q(x). □

**Definition 2.**Define$\ell :\mathcal{P}(M)\times \mathcal{P}(M)\to [0,\mathrm{\pi});(\mu ,\mu *)\mapsto \ell =\ell (\mu ,\mu *)$ by

**Remark 3.**In [2] T. Friedrich remarked that the distance between μ and μ* in$\mathcal{P}(M)$ is given by ℓ = ℓ (μ, μ*).

**Theorem 11.**Let μ and μ

^{*}be arbitrary probability measures in P(M), μ ≠ μ

^{*}. Then, there exists a unique geodesic μ(t), i.e., a curve;$t\in I\subset \mathcal{R}\mapsto \mu (t)\in \mathcal{P}(M)$with μ(0) = μ, $\mu (\mathrm{l})\phantom{\rule{0.3em}{0ex}}=\mu *$, where ℓ = ℓ (μ, μ*) is given by (22) and I is an open interval;

**Proof.**First we will show

_{μ}(τ, τ) = 1, as we compute straightforward

**Assertion 3.**A geodesic joining μ and μ* is unique for μ ≠ μ.

_{μ}tτ and $\tilde{\mu}(t)={\mathrm{exp}}_{\mu}t\tilde{\tau}$be unit speed geodesics satisfying $\mu (0)=\tilde{\mu}(0)=\mu $and $\mu (\ell )=\tilde{\mu}(\ell )=\mu *$. From the latter condition we have by using (15)

_{±}of M by

_{+}∪ M

_{−}. Moreover M

_{+}∩ M

_{−}= ∅. This is because if otherwise, M

_{+}∩ M

_{−}≠ ∅, then at a point x ∈ M

_{+}∩ M

_{−}it holds

_{+}∩ M

_{−}= ∅.

_{+}and M

_{−}are closed in M.

_{−}≠ Ø;. Then, M

_{−}must be a non-empty, proper subset of M. This is because, otherwise if M

_{−}= M is assumed, then, since $\tilde{\tau}$, τ are tangent to P(M) we see, from 0< ℓ < π.

_{−}is a proper subset and hence M

_{+}= M \ M

_{−}is a non-empty closed, but open subset of M. Therefore, since M is connected, M

_{+}= M, namely $\tilde{\tau}(x)=\tau (x)$ for any x ∈ X, from which the assertion is proved.

**Remark 4.**For the ℓ of (22)

^{*}= μ, since ρ is an embedding. Conversely, if μ

^{*}= μ, then$\sqrt{d\mu */d\mu}(x)=1$so$\mathrm{cos}\ell /2={\displaystyle {\int}_{M}\sqrt{d\mu */d\mu}(x)d\mu =1}$ and ℓ = 0.

## 4. Hadamard Manifolds and Barycenter Map

#### 4.1. Hadamard Manifolds and Ideal Boundary

^{n−}

^{1}. For any θ ∈ ∂X Busemann function B

_{θ}normalized at some point and parametrized in θ ∈ ∂X provides a μ-average Busemann function B

_{μ}on X in terms of a probability measure μ on ∂X. Under some geometrical assumptions which X fulfills, B

_{μ}admits a unique critical point so that we have a barycenter map bar : $\mathcal{P}(\partial X)\to X$ by assigning to an arbitrary probability measure μ on ∂X a point in X as its barycenter, a critical point of B

_{μ}.

^{n}(

**R**) are typical examples of Hadamard manifold. Geometrical properties which an Hadamard manifold enjoys are the following;

- Any two points on X can be joined by a unique geodesic.
- Let Δ be a geodesic triangle in X with interior angles α
_{1}, α_{2}, α_{3}and lengths of opposite side, ℓ_{1}, ℓ_{2}, ℓ_{3}. Then, we have a law of cosines;$${\ell}_{3}^{2}\ge {\ell}_{1}^{2}+{\ell}_{2}^{2}-2{\ell}_{1}{\ell}_{2}\mathrm{cos}{\mathrm{\alpha}}_{3}.$$ - The distance function from a fixed point x
_{o}∈ X; ${d}_{{x}_{o}}:X\to \mathbf{R}$, x ↦d(x, x_{o}) is convex, i.e., for any geodesic γ in X; t ↦γ(t) the restricted function ${d}_{{x}_{o}}\circ \mathrm{\gamma}:t\mapsto {d}_{{x}_{o}}(\mathrm{\gamma}(t))$is convex on**R**.

**R**→

**R**is convex, if it satisfies

**R**→ X be unit speed geodesics on X. We say that γ is asymptotically equivalent with σ, denoted by γ ∼ σ, when there exists a constant C > 0 such that d(γ(t), σ(t)) ≤ C for any t ≥ 0. The relation ∼ is an equivalence relation on the space Geo(X) of all oriented, unit speed geodesics on X. The quotient space Geo(X)/∼ is called the ideal boundary of X, denoted by ∂X. An equivalence class represented by γ ∈ Geo(X) is called an asymptotic class, denoted by [γ] or γ(∞). Notice that all geodesics on X are assumed to be of unit speed and oriented.

_{x}X the space of unit tangent vectors at x;

_{x}(tv), t ∈

**R**.

_{1}, θ

_{2}∈ X ∪ ∂X (x ≠ θ

_{1}, x ≠ θ

_{2}) we define ${\angle}_{x}({\theta}_{1},{\theta}_{2})=\angle ({\dot{\mathrm{\gamma}}}_{1}(0),\phantom{\rule{0.3em}{0ex}}{\dot{\mathrm{\gamma}}}_{2}(0))$ angles between a geodesic γ

_{1}from x to θ

_{1}and a geodesic γ

_{2}from x to θ

_{2}. For x ∈ X and θ ∈ ∂X, ε > 0 let C

_{x}(θ, ε) = {θ

_{1}∈ X ∪ ∂X ; θ

_{1}≠ x, ∠ x(θ, θ

_{1}) < ε} be a cone. Further, let T

_{x}(θ, ε) = C

_{x}(θ, ε) \ B(x, r) be a truncated cone (B(x, r) = {y ∈ X|d(y, x) ≤ r} is a closed geodesic ball). Then, a topology generated by open geodesic balls in X and such truncated cones is called a cone topology of X ∪ ∂X. Notice that thus defined cone topology when restricted to ∂X is homeomorphic to the usual topology on S

_{x}X via the mapping β

_{x}. Refer to [31] for the detail.

_{x}be a standard volume measure on S

_{x}X, normalized by (dθ)

_{x}(S

_{x}X) = 1. Through β

_{x}we obtain a measure (β

_{x})

_{#}(dθ)

_{x}on ∂X, denoted by dθ.

#### 4.2. Normalized Busemann Function

**R**→ X be a geodesic on X. Define a function f

_{t}: X →

**R**for t > 0 by

_{t→∞}f

_{t}(x) exists, as we will see in Proposition 2. We write this limit as f

_{∞}(x) and define a function on X; x ↦ f

_{∞}(x), called Busemann function, denoted by B

_{γ}: X → R; x ↦ f

_{∞}(x).

_{γ}is called a horosphere, important in studying geometry of Hadamard manifolds. See [11], for example.

**Example 1.**In a Euclidean space (X, g) = (R

^{n}, g

_{o}) let γ be a geodesic, γ(t) = (t, 0, …, 0). Then, B

_{γ}(x) = −x

^{1}for x = (x

^{1}, …, x

^{n}) ∈

**R**

^{n}from the following

**Example 2.**Busemann function on H

^{n}(R), an n-dimensional real hyperbolic space with standard hyperbolic metric, normalized at o, is given

^{n−}

^{1}.

**Proposition 2.**The functions f

_{t}: X →

**R**, t > 0, introduced above, have a limit lim

_{t→∞}f

_{t}(x) for each x ∈ X.

_{1}), γ(t

_{2})) = t

_{2}− t

_{1}, we see

_{t}, that is |f

_{t}(x)| ≤ d(γ(0), x) for any x ∈ X and t > 0 as follows;

_{t}(x) ≤ d(x, γ(0)).

_{t}(x)|t > 0} is bounded and decreasing and then has a limit as t → ∞.

**Proposition 3.**Let γ and σ be geodesics. If γ ∼ σ, then

_{γ}associated with a geodesic γ, [γ] = θ, gives us a same function on X modulo additive constant. So, let x

_{o}∈ X be an arbitrary point of X as a base point. Then, from non-positive curvature of X there exists a unique geodesic γ such that γ(0) = x

_{o}and [γ] = θ.

**Definition 3.**Let x

_{o}∈ X and θ ∈ ∂X. Let γ :

**R**→ X be a geodesic satisfying γ(0) = x

_{o}and [γ] = θ. The Busemann function B

_{γ}associated to γ is called normalized Busemann function, denoted by B

_{θ}.

**Properties of (normalized) Busemann function:**

- B
_{θ}(x_{o}) = 0 for any θ ∈ ∂X, - B
_{θ}(γ(t)) = −t, t ∈ R, for any θ ∈ ∂X, where γ is a geodesic satisfying γ(0) = x_{o}, [γ] = θ. - Busemann function is Lipschitz continuous;$$|{B}_{\theta}(x)-{B}_{\theta}(y)|\le d(x,y),\phantom{\rule{1em}{0ex}}x,y\in X.$$
- Gradient vector field ∇B
_{θ}satisfies |(∇B_{θ})_{x}| ≡ 1 for any x ∈ X and θ ∈ ∂X. Here (∇B_{θ})_{x}∈ T_{x}X is defined by h(∇B_{θ})_{x}, vi = v(B_{θ}), directional derivative of B_{θ}with respect to v ∈ T_{x}X. An integral curve x(t) of ∇B_{θ}passing through a point x is obtained by x(t) = σ(−t), where σ is a geodesic of σ(0) = x and [σ] = θ. Moreover, for any x ∈ X and any vector v ∈ S_{x}X there exists a θ ∈ ∂X such that v = −(∇B_{θ})_{x}so that β_{x}(v) = θ. - Busemann function is convex (see (26)), since it is a limit of convex functions.
- From (vi), the Hessian of Busemann function (∇dB
_{θ})_{x}: T_{x}× T_{x}→ R is positive semi-definite at any point x ∈ X, i.e., (∇dB_{θ})_{x}(v, v) ≥ 0, for any v ∈ T_{x}X and x ∈ X, and satisfies$${(\nabla d{B}_{\theta})}_{x}({(\nabla {B}_{\theta})}_{x},v)=0,\phantom{\rule{1em}{0ex}}v\in {T}_{x}X.$$^{2}-function f on X the Hessian ∇df is a symmetric bilinear form, defined by$${(\nabla df)}_{x}(u,v)=u\phantom{\rule{0.2em}{0ex}}(Vf)-({\nabla}_{u}V)f,\phantom{\rule{1em}{0ex}}u,v\in {T}_{x}X,\phantom{\rule{0.2em}{0ex}}x\in X$$_{x}X$${(\nabla df)}_{x}(u,u)=\frac{{d}^{2}}{d{t}^{2}}|{}_{t=0}f(\mathrm{\gamma}(t))$$

**Example 3.**On a Euclidean space ∇dB

_{θ}= 0 for any θ ∈ ∂X(due to Example 1).

**Example 4.**On a real hyperbolic space H

^{n}(R), n ≥ 2,

^{*}g = g. An isometry preserves the distance d of X, i.e., d(ϕ(x), ϕ(y)) = d(x, y), x, y ∈ X and transforms a geodesic σ into a new geodesic ϕ ○ σ so that, if σ ∼ γ, then ϕ ○ σ ∼ ϕ ○ γ. Therefore, ϕ induces a transformation $\widehat{\varphi}$ of ∂X, a ∂X-extension of ϕ as

^{−1}of ϕ. $\widehat{\varphi}$ is a homeomorphism of ∂X in terms of cone topology.

**Proposition 4**(Busemann cocycle formula [33]). Any normalized Busemann function enjoys a cocycle formula with respect to an isometry ϕ of X;

**Proof.**Let γ : R → X be a geodesic, γ(0) = x

_{o}, [γ] = θ. Notice that ϕ ○ γ is a geodesic with ϕ ○ γ(0) = ϕ(x

_{o}), which, in general, does not coincide with the base point x

_{o}. For the Busemann function ${B}_{{\varphi}^{-1}\circ \mathrm{\gamma}}(x)$ with respect to a geodesic ϕ

^{−1}○ γ we have

^{−1}○ γ belongs to ${\widehat{\varphi}}^{-}{}^{1}(\theta )$ and (ϕ

^{−1}○ γ)(0) = ϕ

^{−1}(x

_{o}). Let σ be a geodesic such that $[\mathrm{\sigma}]={\widehat{\varphi}}^{-}{}^{1}(\theta ),\mathrm{\sigma}(0)={x}_{o}$ Then, the normalized Busemann function ${B}_{{\widehat{\varphi}}^{-1}(\theta )}$ is given by B

_{σ}. Since ϕ

^{−1}○ γ and σ belong to the same ${\widehat{\varphi}}^{-}{}^{1}(\theta )$, from (29) ${B}_{\mathrm{\sigma}}-{B}_{{\varphi}^{-1}\circ \mathrm{\gamma}}$ is a constant function on X. This constant is given from the above by $({B}_{\mathrm{\sigma}}-{B}_{{\varphi}^{-1}\circ \mathrm{\gamma}})({x}_{o})=-{B}_{{\varphi}^{-1}\circ \mathrm{\gamma}}({x}_{o})=-{B}_{\mathrm{\gamma}}(\varphi ({x}_{o}))$ so that on X

_{θ}is given by B

_{γ}, (28) is obtained from (30). □

_{θ}on X is assumed to be continuous with respect to θ ∈ ∂X for each fixed point x ∈ X. This assumption is guaranteed by a real hyperbolic space. See Example 3. Rank one symmetric spaces of non-compact type and Damek-Ricci spaces satisfy this assumption, as is seen in [25].

**Definition 4**([21]). An Hadamard manifold (X, g) is said to satisfy visibility axiom, if for any distinct ideal point θ, θ

_{1}of ∂X there exists a geodesic γ :

**R**→ X such that γ(+∞) = θ and γ(−∞) = θ

_{1}. Here γ(−∞) ∈ ∂X defined by [γ

^{−}], where γ

^{−}is the geodesic of reversed orientation given by γ

^{−}(t) = γ(−t), t ∈ R.

**Proposition 5.**Let (X, g) be an Hadamard manifold. (X, g) satisfies visibility axiom if and only if, for any θ ∈ ∂X

_{1}≠ θ. Refer to [31] for this.

_{θ}(x) = −∞, if x → θ, from property (i) of normalized Busemann function.

#### 4.3. Average Busemann Function and Barycenter

_{θ}(x) is continuous with respect to every θ.

^{n−}

^{1}, n = dim X and dθ a normalized standard measure on ∂X.

^{0}and positive;

**Definition 5.**Let.$\mu \in \mathcal{P}(M)$ Then, a function B

_{μ}: X → R, called μ-average Busemann function, is defined by

- For any $\mu \in \mathcal{P}(M)$ each B
_{μ}is convex on X and B_{μ}(x_{o}) = 0. - B
_{μ}(γ(t)) → +∞, as t → ∞, where γ :**R**→ X is an arbitrary geodesic in X (Theorem 12). - B
_{μ}is Lipschitz continuous, in fact, |B_{μ}(x) − B_{μ}(y)| ≤ |d(x, y)| for x, y ∈ X. - The gradient vector field ∇B
_{μ}is defined on X as$${(\nabla d{\mathrm{B}}_{\mu})}_{x}={\displaystyle {\int}_{\partial X}{(\nabla {B}_{\theta})}_{x}d\mu (\theta ),\phantom{\rule{1em}{0ex}}}x\in X$$_{μ})_{x}| ≤ 1, x ∈ X. - the Hessian ∇dB
_{μ}can be defined as μ-average Hessian;$${(\nabla d{\mathrm{B}}_{\mu})}_{x}(u,v)={\displaystyle {\int}_{\partial X}{(\nabla d{B}_{\theta})}_{x}(u,v)d\mu (\theta ),\phantom{\rule{1em}{0ex}}u,v\in {T}_{x}X\phantom{\rule{0.5em}{0ex}}}x\in X,$$_{θ}, the Laplacian of B_{θ}together with d(ΔB_{θ}) are uniformly bounded with respect to x ∈ X and θ ∈ ∂X. Here Δf = −trace ∇df for a C^{2}-function f on X. This is derived from Bochner formula (see [23]). If (X, g) is asymptotically harmonic ([34]), i.e., ΔB_{θ}≡ c for any θ, and of bounded Ricci curvature, the average Hessian ∇dB_{μ}, $\mu \in \mathcal{P}(M)$ is defined.

**Definition 6.**Let$\mu \in \mathcal{P}(M)$. A critical point of μ-average Busemann function B

_{μ}is called a barycenter of μ.

^{1}-function f : X →

**R**, y ∈ X is called a critical point of f, if one of the following equivalent conditions holds;

- the differential of f at y vanishes along all directional vector, i.e.,$$\frac{d}{dt}{|}_{t=0}f(x(t))=0$$
^{1}-curve x(t) of x(0) = y, - the one-form df, or the gradient vector field ∇f vanishes at y.

**Observation.**For $\mu =p(\theta )d\theta \in \mathcal{P}(\partial X)$, x ∈ X is a barycenter of μ if and only if (dB

_{μ})

_{x}(u) = 0 for any u ∈ T

_{x}X, which is equivalent to stating that a measure τ defined by τ = (dB

_{θ})

_{x}(u) dμ = h(∇B

_{θ})

_{x}, ui p(θ) dθ is a tangent vector to $\mathcal{P}(\partial X)$ at μ for each u ∈ T

_{x}X.

**Theorem 12.**If, as is assumed, an Hadamard manifold (X, g) satisfies visibility axiom and Busemann function is continuous with respect to any θ ∈ ∂X. Then, every$\mu \in \mathcal{P}(M)$ admits a barycenter.

**Proof.**This theorem is proved by Besson, Courtois and Gallot in [8] by showing B

_{μ}(γ(t)) → ∞ as t → +∞ along any geodesic γ of X. However, they assume that all probability measures on ∂X are without atom and an Hadamard manifold (X, g) is of special type, i.e., a rank one symmetric space of non-compact type. We restrict the space of probability measures as $\mathcal{P}(\partial X)$. However, we relax the assumptions concerning an Hadamard manifold (X, g) and then, assume only that (X, g) satisfies visibility axiom and Busemann function is continuous with respect to any θ ∈ ∂X.

_{C}= {y ∈ X ; B

_{μ}(y) ≤ C}. A

_{C}is a convex set and x

_{o}∈ A

_{C}, since B

_{μ}is convex and B

_{μ}(x

_{o}) = 0. Note A

_{C}contains a geodesic ball {y ∈ X ; d(y, x

_{o}) ≤ C/2} since B

_{μ}is Lipschitz. Let $\mu \in \mathcal{P}(M)$ and γ be a geodesic satisfying γ(0) = x

_{o}and [γ] = θ. Then it is possible to verify lim

_{t→}

_{+}

_{∞}B

_{μ}(γ(t)) = +∞ in the following steps;

**Step I.**Since Busemann function B

_{θ}is convex and B

_{θ}(x

_{0}) = 0 for any θ, we have

_{1}/t, then, 0 ≤ a ≤ 1 and we have t

_{1}= (1 − a) 0 + at. The Convex function B

_{θ}(γ(t)) fulfills

**Step II.**Fix t

_{1}> 0 of Step I. Take an arbitrary θ

_{0}∈ ∂X and fix it. Let γ be a geodesic of γ(0) = x

_{0}and [γ] = θ

_{0}. For any t > 0 set

_{1}∈ (0, ∞) such that $\mu ({J}_{{\theta}_{0}}({t}_{1}))<1$, as follows.

_{θ}(x) is continuous with respect to θ, the set ${J}_{{\theta}_{0}}(t)$ is compact in ∂X. We see ${\theta}_{0}\in {J}_{{\theta}_{0}}(t)$. From (32) it holds

_{0}lim

_{t→∞}B

_{θ}(γ(t)) = +∞. Moreover, from (33) for the μ we find

_{1}∈ (0, ∞) such that $\mu ({J}_{{\theta}_{0}}({t}_{1}))<1$. Here, notice $\mu ({J}_{{\theta}_{0}}(t))\le \mu ({J}_{{\theta}_{0}}({t}_{1}))<1$ for any t ≥ t

_{1}> 0.

**Step III.**Let K be a compact subset of $\partial X\backslash {J}_{{\theta}_{0}}({t}_{1})$ satisfying μ(K) > 0. It is possible to choose such a K. Then, from (33) it holds for any t ≥ t

_{1}that $K\subset \partial X\backslash {J}_{{\theta}_{0}}(t)$, and B

_{θ}(γ(t)) ≥ 0 for any $\theta \in \partial X\backslash {J}_{{\theta}_{0}}(t)$. So,

_{θ}(γ(t

_{1})), as a continuous function of θ, is bounded with respect to θ. Therefore, the above is written as

_{C}is bounded and hence is compact. Therefore, B

_{μ}admits a minimal point x ∈ X, namely, x is a barycenter of μ. □

**Proposition 6.**Let (X, g) be an Hadamard manifold of bounded Ricci curvature. If (X, g) is asymptotically harmonic, then the following holds; If there exists${\mu}_{0}\in \mathcal{P}(M)$ such that μ-average Hessian$\nabla d{\mathcal{B}}_{{\mu}_{0}}$is positive definite at every point in X, then, for any μ ∈ P(M) μ-average Hessian ∇dB

_{μ}is also positive definite at every point in X.

**Proof.**Let x ∈ X and u ∈ T

_{x}X. Then, for a geodesic γ in X, γ(0) = x, $\dot{\mathrm{\gamma}}(0)=u$we have

**Theorem 13.**Let (X, g) be an Hadamard manifold satisfying the above assumptions. Then, any$\mu \in \mathcal{P}(M)$ admits a unique barycenter.

_{1}, y

_{2}, y

_{1}≠ y

_{2}, then f(t) := B

_{μ}(γ(t)) along a geodesic γ :

**R**→ X joining y

_{1}and y

_{2}satisfies f′(0) = f′ (d) = 0 (d = d(y

_{1}, y

_{2})) and f″ (t) > 0, t ∈ [0, d] because of the positive definiteness of μ-average Hessian (35). So, f(t) must be constant along γ. This contradicts property (ii) of μ-average Busemann function. Hence uniqueness is proved.

**Proposition 7**(average Busemann cocycle formula). Let ϕ be an isometry of an Hadamard manifold (X, g). Then for any$\mu \in \mathcal{P}(\partial X)$

**Proof.**Integrate the Busemann cocycle formula (28)

^{−1}with respect to a measure μ. We then get (36). □

**Example 5.**The standard measure dθ has bar(dθ) = x

_{o}, the base point as its barycenter. In fact, we observe

^{n−}

^{1}which is described as${\sum}_{i=1}^{n}{({\theta}^{i})}^{2}=1$ with respect to the standard coordinates θ = (θ

^{1}, …, θ

^{n}) ∈

**R**

^{n};

#### 4.4. Barycenter Map

^{0}-function f on ∂X, find a function u on X which satisfies the Laplace equation Δu = 0 and the boundary condition lim

_{x→θ}u(x) = f(θ) for θ ∈ ∂X.

**Definition 7**([3,35]). A function P

_{θ}(x) = P (x, θ) on X is called a Poisson kernel, normalized at x

_{o}, for θ ∈ ∂X if it satisfies

- ΔP (x, θ) = 0 and P (x, θ) > 0 for any x ∈ X and θ ∈ ∂X.
- P (x
_{o}, θ) = 1 for any θ ∈ ∂X. - for any θ ∈ ∂X, P (x, θ) ∈ C
^{0}(X ∪ ∂X \ {θ}) as an extension function on X ∪ ∂X and lim_{x→θ}_{1}P (x, θ) = 0 for θ_{1}≠ θ.

**Example 6.**On a real hyperbolic space H

^{n}(

**R**) of standard hyperbolic metric of Poicaré ball model, the Poisson kernel is given by

**Example 7.**The Poisson integral formula, well known in potential theory, is for a bounded harmonic function h = h(z), z = re

^{iφ}∈ {z ∈ C||z| ≤ 1}

^{1}. The kernel function (1 − r

^{2})/(1 − 2r cos(ϕ − θ) + r

^{2}) is just the Poisson kernel P (z, θ) = (1 − |z|

^{2})/|z − θ|

^{2}of the hyperbolic plane H

^{2}(

**R**). See, for example [20].

**Definition 8.**A Poisson kernel on an Hadamard manifold (X, g) is called Busemann-Poisson kernel, when it has the following form

**Remark 5.**For volume entropy refer to [8] in which the following theorem, Theorem of Manning, ([7]) is cited; if Q

_{top}denotes the topological entropy of a compact Riemannian manifold Y of non-positive curvature, then one has

- Q(Ỹ) ≤ Q
_{top}(Y), - Q(Ỹ) = Q
_{top}(Y), provided the curvature of Y is negative or zero.Here, Ỹ is the universal covering space of Y and the topological entropy Q_{top}(Y) is defined by$${Q}_{top}(Y)=\underset{R\to \infty}{\mathrm{lim}}\frac{1}{R}\mathrm{log}(\#\{\mathrm{\gamma}|\ell (\mathrm{\gamma})\le R\}),$$**R**.

^{n}(

**R**) of standard hyperbolic metric.

**Remark 6.**Any Damek-Ricci space admits a Busemann-Poisson kernel (refer to [4]). See also [8] for a rank one symmetric space of non-compact type which is just a member of Damek-Ricci spaces, as observed by using Iwasawa decomposition of isometry groups.

**Theorem 14.**Let (X, g) be an Hadamard manifold satisfying the assumptions in Theorem 12 and Proposition 6. If (X, g) admits a Busemann-Poisson kernel, then, for${\mu}_{x}:=P(x,\theta )d\theta \in \mathcal{P}(\partial X)$

- bar(μ
_{x}) = x for any x ∈ X and - at any point y ∈ X, (∇dB
_{μx})_{y}is positive definite.

**Definition 9.**Let$\mu =p(\theta )d\theta \in \mathcal{P}(\partial X)$and x ∈ X be a barycenter of μ. We define a linear map

**Proof of Theorem**14. (i) Let u ∈ T

_{x}X and x(t) a C

^{1}-curve in X such that x(0) = x, $\dot{x}(0)=u$. Differentiate ${\int}_{\partial X}P(x(t),\theta )d\theta \equiv 1$ as

_{x}.

**Assertion 4.**The measure μ

_{x}satisfies

_{θ})

_{y}(·,·) is positive semi-definite,

_{y}X, where C = inf

_{θ}

_{∈}

_{∂X}P (x, θ)/P (y, θ) > 0. □

**Theorem 15.**The barycenter map bar : $\mathcal{P}(\partial X)\to X$ gives a projection of a fibre space whose total space is$\mathcal{P}(\partial X)$ and base space is X with fibres bar

^{−1}(x) over x ∈ X. In fact, let x ∈ X and μ ∈ bar

^{−1}(x). Then

_{μ}bar

^{−1}(x) and the horizontal subspace$\mathrm{Im}{v}_{x}^{\mu}$ with respect to Fisher metric G

_{μ}.

^{−1}(x)} distributes a normal bundle to each fibre bar

^{−1}(x), x ∈ X. Notice that bar

^{−1}(x) is path-connected, since, for μ, μ

_{1}∈ bar

^{−1}(x) (1 − t)μ + tμ

_{1}, 0 ≤ t ≤ 1, also belongs to bar

^{−1}(x).

_{μ}bar

^{−1}(x) is orthogonal to ${N}_{\mu}=\mathrm{Im}{v}_{x}^{\mu}$. Let u ∈ T

_{x}X and τ ∈ T

_{μ}bar

^{−1}(x) and take μ(t) = μ + tτ for sufficiently small |t|. Then, μ (t) ∈ bar

^{−1}(x). So, for a sufficiently small |t|, we have

_{μ}bar

^{−1}(x) and ${N}_{\mu}=\mathrm{Im}{v}_{x}^{\mu}$.

#### 4.5. Fibres bar^{−1}(x) and Geodesics

^{−1}(x).

**Theorem 16.**Let (X, g) be an Hadamard manifold satisfying the assumptions in Theorem 12 and Proposition 6, and admitting a Busemann-Poisson kernel.

^{−1}(x) and τ ∈ T

_{μ}bar

^{−1}(x), |τ|

_{G,μ}= 1. Then a geodesic μ(t) = exp

_{μ}tτ entirely belongs to bar

^{−1}(x) for any t at which μ(t) is well-defined, if and only if τ fulfills H

_{μ}(τ, τ) = 0.

^{−1}(x) of the ambient space $\mathcal{P}(\partial X)$ (see Equation (5) in Section 2).

**Proof.**From Theorem 10, Section 3 the geodesic μ(t) is given by

^{−1}(x) for all t if and only if for any u ∈ T

_{x}X

^{−1}(x) and τ ∈ T

_{μ}bar

^{−1}(x), this is equivalent to

_{μ}tτ belongs to the fibre bar

^{−1}(x) by following reversely the above argument.

**Theorem 17.**Let μ, μ* ∈ bar

^{−1}(x), x ∈ X (μ ≠ μ*). Then, a geodesic μ(t) joining μ and μ* lies entirely on bar

^{−1}(x) if and only if

**Proof.**The geodesic μ(t) joining μ and μ* is written from Theorem 11 by exp

_{μ}tτ of an initial vector

^{−1}(x) if and only if the following conditions hold, that is, τ is tangent to bar

^{−1}(x), namely,

_{x}X, and that

_{x}X. This condition is also (40), so we get Theorem 17.

**Example 8.**Let μ = dθ. Then, bar(dθ) = x

_{o}, as seen in Example 5. We exhibit tangent vectors τ, τ

_{1}at dθ satisfying H(τ; τ) = 0, whereas H(τ

_{1}; τ

_{1}) ≠ 0, as follows;

- Identify ∂X with ${S}_{{x}_{o}}X\cong {S}^{n-1}$ via ${\mathit{\beta}}_{{x}_{o}}$, and dθ with ${(d\theta )}_{{x}_{o}}$. Choose on S
^{n−1}a function q = q(θ) = θ^{i}θ^{j}, i ≠ j and define τ = q(θ) dθ as a measure on ∂X. Then, $\tau \in {T}_{d\theta}\mathcal{P}(\partial X)$. Moreover, τ ∈ T_{dθ}bar^{−1}(x_{o}), since ${G}_{d\theta}({v}_{{x}_{o}}^{d\theta}(u),\tau )=0$ for any $u\in {T}_{{x}_{o}}X$and H(τ, τ) = 0. These are directly from the integral formulae; ${\int}_{{S}^{n-1}}{\theta}^{i}{\theta}^{j}{\theta}^{k}}{(d\theta )}_{{x}_{o}}=0$, ${\int}_{{S}^{n-1}}{({\theta}^{i}{\theta}^{j})}^{2}{\theta}^{k}}{(d\theta )}_{{x}_{o}}=0$ for any k = 1, …, n. By normalizing τ′ = τ/|τ|_{G}in terms of G, from Theorem 16 γ(t) = exp_{dθ}tτ′ gives a geodesic lying on bar^{−1}(x_{o}). - Let q
_{1}= q_{1}(θ) is a function on S^{n−1}, n ≥ 3, defined by q_{1}(θ) = θ^{1}θ^{2}θ^{3}+θ^{2}θ^{3}and set τ_{1}= q_{1}(θ) dθ. Then $({\mathrm{\gamma}}_{1})(t)={\mathrm{exp}}_{d\theta}t{{\tau}^{\prime}}_{1}$, ${{\tau}^{\prime}}_{1}={\tau}_{1}/|{\tau}_{1}{|}_{G}$ is a geodesic being not completely on the fibre bar^{−1}(x_{o}).

## 5. Barycentrically Associated Maps

**Theorem 18**([8]). For any isometry ϕ of (X, g), we have

**Proof.**Let $y=\text{bar(}{\widehat{\varphi}}_{\#}\mu \text{)}$. Then ${(d{\mathcal{B}}_{{\widehat{\varphi}}_{\#}\mu})}_{y}(u)=0$ for any u ∈ T

_{y}X, namely, due to (36) ϕ

^{−1}y turns out to be a critical point of B

_{μ}, that is, y = ϕ(bar(μ)), so (44) is obtained. □

**Definition 10.**Let Φ : ∂X → ∂X be a homeomorphism of ∂X. Then, a bijective map ϕ : X → X is said to be barycentrically associated to Φ, if Φ and ϕ satisfy the relation bar ◯ Φ = ϕ ◯ bar, that is, bar(Φ(μ)) = ϕ(bar(μ)) for any$\mu \in \mathcal{P}(\partial X)$.

**Proof of Theorem 6.**From the statement of the theorem, diagram (6) asserts for any x ∈ X, i.e.,

_{x}= Σ(x), where $\mathrm{\Sigma}:X\to \mathcal{P}(\partial X)$ is a cross section whose existence is assumed in the theorem. We write (46) as

_{*})

_{x})* of (φ

_{*})

_{x}, we write the above as

_{x}X be a unit tangent vector at φx and choose θ ∈ ∂X such that (∇B

_{θ})

_{φx}= v so that the above is written as ${({\phi}_{*})}_{x}^{*}v={(\nabla {B}_{{\mathrm{\Phi}}^{-1}\theta})}_{x}$ and thus from property (v) in Section 4.2 it is concluded that $|{({\phi}_{*})}_{x}^{*}v|=|{(\nabla {B}_{{\mathrm{\Phi}}^{-1}\theta})}_{x}|=1$ which implies that ${({\phi}_{*})}_{x}^{*}$ and consequently (φ

_{*})

_{x}is a linear isometry. Since x ∈ X is arbitrary, φ turns out to be an isometry of (X, g).

_{x}X, x ∈ X

^{−1}θ. Then, from the visibility axiom (see Proposition 5) ${B}_{{\widehat{\phi}}^{-1}\theta}(x)-{B}_{{\mathrm{\Phi}}^{-1}\theta}(x)\to -\infty $ contradicting that C is constant.

## Author Contributions

## Conflicts of Interest

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Itoh, M.; Satoh, H.
Geometry of Fisher Information Metric and the Barycenter Map. *Entropy* **2015**, *17*, 1814-1849.
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**AMA Style**

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Geometry of Fisher Information Metric and the Barycenter Map. *Entropy*. 2015; 17(4):1814-1849.
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Itoh, Mitsuhiro, and Hiroyasu Satoh.
2015. "Geometry of Fisher Information Metric and the Barycenter Map" *Entropy* 17, no. 4: 1814-1849.
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