Symplectic and Kähler Structures on Statistical Manifolds Induced from Divergence Functions

28/08/2013
Auteurs : Jun Zhang
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Symplectic and Kähler Structures on Statistical Manifolds Induced from Divergence Functions

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Symplectic and K¨ahler Structures on Statistical Manifolds Induced from Divergence Functions Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu) (In collaboration with Fubo Li, Sichuan University, China) Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 1. Divergence Functions and Dual Geometry Definition: Divergence Function Let V ⊆ Rn denote a local chart for a manifold M. A divergence function D : V × V → R≥0 on V , or on M through pullback, is a smooth function (differentiable up to third order) which satisfies (i) D(x, y) ≥ 0 ∀x, y ∈ V with equality holding iff if x = y; (ii) Di (x, x) = D,j (x, x) = 0, ∀i, j ∈ {1, · · · , n}; (iii) −Di,j (x, x) is positive definite. Here Di (x, y) = ∂xi D(x, y), D,i (x, y) = ∂yi D(x, y), Di,j (x, y) = ∂xi ∂yj D(x, y), etc. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Statistical Manifold from a Divergence Function It is well-known that a statistical structure {M, g, Γ, Γ∗ } can be induced from a divergence function. Lemma (Eguchi, 1983; 1992) A divergence function induces a Riemannian metric g and a pair of torsion-free conjugate connections Γ, Γ∗ gij (x) = −Di,j (x, x); Γij,k(x) = −Dij,k(x, x) ; Γ∗ ij,k(x) = −Dk,ij (x, x) . The proof is through verifying that i) gij , Γij,k, Γ∗ ij,k as given above satisfy the definition for conjugate connection; ii) under arbitrary coordinate transform, these quantities behave properly as desired. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 2. Convex-Based Divergence Functions Let Φ : V ⊆ Rn → R be a strictly convex function defined on a convex set S = int dom(Φ) ⊆ Rn , i.e., for any two points x ∈ S, y ∈ S and any real number α ∈ (−1, 1), 1 − α 2 Φ(x) + 1 + α 2 Φ(y) − Φ 1 − α 2 x + 1 + α 2 y ≥ 0. As the inequality sign is reversed when |α| > 1, we define DΦ-Divergence Introduced in Zhang (2004) D (α) Φ (x, y) = 4 1 − α2 1 − α 2 Φ(x) + 1 + α 2 Φ(y) − Φ 1 − α 2 x + 1 + α 2 y , with D (α) Φ (x, y) = D (−α) Φ (y, x). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 2. Convex-Based Divergence Functions Let Φ : V ⊆ Rn → R be a strictly convex function defined on a convex set S = int dom(Φ) ⊆ Rn , i.e., for any two points x ∈ S, y ∈ S and any real number α ∈ (−1, 1), 1 − α 2 Φ(x) + 1 + α 2 Φ(y) − Φ 1 − α 2 x + 1 + α 2 y ≥ 0. As the inequality sign is reversed when |α| > 1, we define DΦ-Divergence Introduced in Zhang (2004) D (α) Φ (x, y) = 4 1 − α2 1 − α 2 Φ(x) + 1 + α 2 Φ(y) − Φ 1 − α 2 x + 1 + α 2 y , with D (α) Φ (x, y) = D (−α) Φ (y, x). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Bregman (Canonical) Divergence when α = ±1 It can be shown that D (1) Φ (x, y) = D (−1) Φ (y, x) = BΦ(x, y), D (−1) Φ (x, y) = D (1) Φ (y, x) = BΦ(y, x), where BΦ is the Bregman divergence BΦ(x, y) = Φ(x) − Φ(y) − x − y, ∂Φ(y) . Using the convex conjugate Φ of Φ, and dual variables: u = (∂Φ)(x) = (∂Φ)−1 (x) ←→ x = (∂Φ)(u) = (∂Φ)−1 (u), Bregman divergence becomes the “canonical” divergence AΦ : V × V → R≥0 (and AΦ : V × V → R≥0) AΦ(x, v) = Φ(x) + Φ(v) − x, v = AΦ(v, x). via BΦ(x, (∂Φ)−1 (v)) = AΦ(x, v) = BΦ((∂Φ)(x), v). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Bregman (Canonical) Divergence when α = ±1 It can be shown that D (1) Φ (x, y) = D (−1) Φ (y, x) = BΦ(x, y), D (−1) Φ (x, y) = D (1) Φ (y, x) = BΦ(y, x), where BΦ is the Bregman divergence BΦ(x, y) = Φ(x) − Φ(y) − x − y, ∂Φ(y) . Using the convex conjugate Φ of Φ, and dual variables: u = (∂Φ)(x) = (∂Φ)−1 (x) ←→ x = (∂Φ)(u) = (∂Φ)−1 (u), Bregman divergence becomes the “canonical” divergence AΦ : V × V → R≥0 (and AΦ : V × V → R≥0) AΦ(x, v) = Φ(x) + Φ(v) − x, v = AΦ(v, x). via BΦ(x, (∂Φ)−1 (v)) = AΦ(x, v) = BΦ((∂Φ)(x), v). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 3. α-Hessian Manifold A manifold {M, g, Γ, Γ∗ } with Riemannian metric g and a pair of torsion-free, conjugate connections Γ, Γ∗ is said to be a “statistical manifold”. Statistical manifolds admit a one-parameter family of affine connections Γ(α) , called “α-connections” (α ∈ R): Γ(α) = 1 + α 2 Γ + 1 − α 2 Γ∗ . Obviously, (Γ(α) )∗ = Γ(−α) , Γ(0) = Γ. When Γ and Γ∗ are dually flat, the manifold {M, g, Γ, Γ∗ } is called a “Hessian manifold”. In such a case, Γ(α) is called “α-transitively flat” (Uohashi, 2002), and {M, g, Γ(α) , Γ(−α) } is hereby called an “α-Hessian manifold”. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 3. α-Hessian Manifold A manifold {M, g, Γ, Γ∗ } with Riemannian metric g and a pair of torsion-free, conjugate connections Γ, Γ∗ is said to be a “statistical manifold”. Statistical manifolds admit a one-parameter family of affine connections Γ(α) , called “α-connections” (α ∈ R): Γ(α) = 1 + α 2 Γ + 1 − α 2 Γ∗ . Obviously, (Γ(α) )∗ = Γ(−α) , Γ(0) = Γ. When Γ and Γ∗ are dually flat, the manifold {M, g, Γ, Γ∗ } is called a “Hessian manifold”. In such a case, Γ(α) is called “α-transitively flat” (Uohashi, 2002), and {M, g, Γ(α) , Γ(−α) } is hereby called an “α-Hessian manifold”. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 3. α-Hessian Manifold A manifold {M, g, Γ, Γ∗ } with Riemannian metric g and a pair of torsion-free, conjugate connections Γ, Γ∗ is said to be a “statistical manifold”. Statistical manifolds admit a one-parameter family of affine connections Γ(α) , called “α-connections” (α ∈ R): Γ(α) = 1 + α 2 Γ + 1 − α 2 Γ∗ . Obviously, (Γ(α) )∗ = Γ(−α) , Γ(0) = Γ. When Γ and Γ∗ are dually flat, the manifold {M, g, Γ, Γ∗ } is called a “Hessian manifold”. In such a case, Γ(α) is called “α-transitively flat” (Uohashi, 2002), and {M, g, Γ(α) , Γ(−α) } is hereby called an “α-Hessian manifold”. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu α-Hessian Structure Induced from DΦ-Divergence Applying Eguchi relation to the divergence function D (α) Φ (x, y) results in an α-Hessian structure of M. Theorem (Zhang, 2004) The manifold {M, g(x), Γ(α) (x), Γ(−α) (x)} associated with D (α) Φ (x, y) is given by gij (x) = Φij and Γ (α) ij,k(x) = 1 − α 2 Φijk, Γ ∗(α) ij,k (x) = 1 + α 2 Φijk. Here, Φij , Φijk denote, respectively, second and third partial derivatives of Φ(x) : Φij = ∂2Φ(x) ∂xi ∂xj , Φijk = ∂3Φ(x) ∂xi ∂xj ∂xk . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu α-Hessian Structure: Curvature and Volume Form Theorem (Zhang, 2004; Zhang, 2007; Zhang and Matsuzoe, 2009) For α-Hessian manifold {M, g(x), Γ(α) (x), Γ(−α) (x)}, 1 the α-Riemann curvature tensor is given by: R (α) µνij (x) = 1 − α2 4 l,k (ΦilνΦjkµ − ΦilµΦjkν)Ψlk = R ∗(α) ijµν (x), with Ψij being the matrix inverse of Φij ; 2 the α-parallel volume form is given by Ω(α) (x) = det[Φij (x)] 1−α 2 . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu α-Hessian Structure: Biorthogonal Coordinates Consider coordinate transform x → u, ∂i ≡ ∂ ∂ui = l ∂xl ∂ui ∂ ∂xl = l Fli ∂l where the Jacobian matrix F is given by Fij (x) = ∂ui ∂xj , Fij (u) = ∂xi ∂uj , l Fil Flj = δl k where δj i is Kronecker delta. If the new coordinate system u = [u1, · · · , un] is such that Fij (x) = gij (x), then the x-coordinate system and the u-coordinate system are said to be “biorthogonal” to each other since g(∂i , ∂j ) = g(∂i , l Flj ∂l ) = l Flj g(∂i , ∂l ) = l Flj gil = δj i . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu α-Hessian Structure: Biorthogonal Coordinates Consider coordinate transform x → u, ∂i ≡ ∂ ∂ui = l ∂xl ∂ui ∂ ∂xl = l Fli ∂l where the Jacobian matrix F is given by Fij (x) = ∂ui ∂xj , Fij (u) = ∂xi ∂uj , l Fil Flj = δl k where δj i is Kronecker delta. If the new coordinate system u = [u1, · · · , un] is such that Fij (x) = gij (x), then the x-coordinate system and the u-coordinate system are said to be “biorthogonal” to each other since g(∂i , ∂j ) = g(∂i , l Flj ∂l ) = l Flj g(∂i , ∂l ) = l Flj gil = δj i . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Biorthogonal Coordinates (cont) In such a case, denote gij (u) = g(∂i , ∂j ), which equals Fij (u), the Jacobian of the inverse coordinate transform u → x. Let (the unconventional notation) Γrs t , Γ∗rs t and Γrs,t , Γ∗rs,t denote the contravariant and covariant versions, respecrively, under the u-coordinate system: ∂r ∂s = Γrs t ∂t ; ∗ ∂r ∂s = Γ∗rs t ∂t . Γij,k (u) = g( ∂i ∂j , ∂k ), Γ∗ij,k (u) = g( ∗ ∂i ∂j , ∂k ). Then, gik (u)gkj (x) = δi j ; Γ∗ ts r (u) = −gjs (u)Γt jr (x); Γ∗ts,r (u) = −gir (u)gjs (u)gkt (u)Γij,k(x). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Biorthogonal Coordinates (cont) In such a case, denote gij (u) = g(∂i , ∂j ), which equals Fij (u), the Jacobian of the inverse coordinate transform u → x. Let (the unconventional notation) Γrs t , Γ∗rs t and Γrs,t , Γ∗rs,t denote the contravariant and covariant versions, respecrively, under the u-coordinate system: ∂r ∂s = Γrs t ∂t ; ∗ ∂r ∂s = Γ∗rs t ∂t . Γij,k (u) = g( ∂i ∂j , ∂k ), Γ∗ij,k (u) = g( ∗ ∂i ∂j , ∂k ). Then, gik (u)gkj (x) = δi j ; Γ∗ ts r (u) = −gjs (u)Γt jr (x); Γ∗ts,r (u) = −gir (u)gjs (u)gkt (u)Γij,k(x). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Biorthogonal Coordinates (cont) In such a case, denote gij (u) = g(∂i , ∂j ), which equals Fij (u), the Jacobian of the inverse coordinate transform u → x. Let (the unconventional notation) Γrs t , Γ∗rs t and Γrs,t , Γ∗rs,t denote the contravariant and covariant versions, respecrively, under the u-coordinate system: ∂r ∂s = Γrs t ∂t ; ∗ ∂r ∂s = Γ∗rs t ∂t . Γij,k (u) = g( ∂i ∂j , ∂k ), Γ∗ij,k (u) = g( ∗ ∂i ∂j , ∂k ). Then, gik (u)gkj (x) = δi j ; Γ∗ ts r (u) = −gjs (u)Γt jr (x); Γ∗ts,r (u) = −gir (u)gjs (u)gkt (u)Γij,k(x). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Existence of Biorthogonal Coordinates Theorem (Shima, 2007; Zhang and Matsuzoe, 2009) The following are equivalent: 1 {M, g} admits biorthogonal coordinates; 2 ∂kgij is totally symmetric, i.e., ∂gij (x) ∂xk = ∂gik(x) ∂xj ; 3 There exists a smooth convex function Φ such that gij (x) = ∂2 Φ(x) ∂xi ∂xj . Furthermore, gij (u) = ∂2Φ(u) ∂ui ∂uj . 4 M is a Hessian manifold, i.e., it admits a pair of dually-flat Γ, Γ∗ such that Γk ij (x) = 0, Γ∗ij k (u) = 0. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 4. Symplectic Structure and Divergence Function It is known that: 1 T ∗ M has a canonical sympletic structure (independent of the Riemannian structure on M); 2 T M has a K¨ahler structure (i.e., complex structure compatible with Riemannian metric) when M is a Hessian manifold. Question: What structure do we have on M × M, the product manifold? A divergence function D, previously viewed a bi-variable function on M (of dim n), can also be viewed as a single-variable function on M × M (of dim 2n) that vanishes along the diagonal (of dim n) ∆M ⊂ M × M. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 4. Symplectic Structure and Divergence Function It is known that: 1 T ∗ M has a canonical sympletic structure (independent of the Riemannian structure on M); 2 T M has a K¨ahler structure (i.e., complex structure compatible with Riemannian metric) when M is a Hessian manifold. Question: What structure do we have on M × M, the product manifold? A divergence function D, previously viewed a bi-variable function on M (of dim n), can also be viewed as a single-variable function on M × M (of dim 2n) that vanishes along the diagonal (of dim n) ∆M ⊂ M × M. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Symplectic Form by Barndorff-Nielsen Construction by (Barndorff-Nielsen and Jupp, 1997) A symplectic form on M × M associated with any D is defined as ωD(x, y) = −Di,j (x, y)dxi ∧ dyj Example. Bregman divergence BΦ induces Φij dxi ∧ dyj . Fixing the value of y in (the 2nd slot of) M × M results in an n-dim submanifold denoted as Mx ≡ M × {y}. Likewise, fixing the value of x results in My ≡ {x} × M. The canonical symplectic forms ωx , ωy on T ∗ Mx , T ∗ My are ωx = dxi ∧ dξi . ωy = dyi ∧ dηi . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Symplectic Form by Barndorff-Nielsen Construction by (Barndorff-Nielsen and Jupp, 1997) A symplectic form on M × M associated with any D is defined as ωD(x, y) = −Di,j (x, y)dxi ∧ dyj Example. Bregman divergence BΦ induces Φij dxi ∧ dyj . Fixing the value of y in (the 2nd slot of) M × M results in an n-dim submanifold denoted as Mx ≡ M × {y}. Likewise, fixing the value of x results in My ≡ {x} × M. The canonical symplectic forms ωx , ωy on T ∗ Mx , T ∗ My are ωx = dxi ∧ dξi . ωy = dyi ∧ dηi . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Symplectic Form by Barndorff-Nielsen Construction by (Barndorff-Nielsen and Jupp, 1997) A symplectic form on M × M associated with any D is defined as ωD(x, y) = −Di,j (x, y)dxi ∧ dyj Example. Bregman divergence BΦ induces Φij dxi ∧ dyj . Fixing the value of y in (the 2nd slot of) M × M results in an n-dim submanifold denoted as Mx ≡ M × {y}. Likewise, fixing the value of x results in My ≡ {x} × M. The canonical symplectic forms ωx , ωy on T ∗ Mx , T ∗ My are ωx = dxi ∧ dξi . ωy = dyi ∧ dηi . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Relation to Canonical Symplectic Form Now define a map LD from M × M → T ∗ Mx , (x, y) → (x, ξ) LD : (x, y) → (x, Di (x, y)dxi ). The map LD is a diffeomorphism since its Jacobian δij Dij 0 Di,j is nondegenerate in a neighborhood U of ∆M. We likewise define RD : M × M → T ∗ My RD : (x, y) → (D,i (x, y)dyi , y). Lemma (Pullback of canonical symplectic form) ωD defined on M × M is the pullback by the maps LD and RD of the canonical ωx defined on T ∗ Mx and ωy on T ∗ My : L∗ D ωx = Di,j (x, y)dxi ∧ dyj = −ωD, R∗ Dωy = −Di,j (x, y)dxi ∧ dyj = ωD. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Relation to Canonical Symplectic Form Now define a map LD from M × M → T ∗ Mx , (x, y) → (x, ξ) LD : (x, y) → (x, Di (x, y)dxi ). The map LD is a diffeomorphism since its Jacobian δij Dij 0 Di,j is nondegenerate in a neighborhood U of ∆M. We likewise define RD : M × M → T ∗ My RD : (x, y) → (D,i (x, y)dyi , y). Lemma (Pullback of canonical symplectic form) ωD defined on M × M is the pullback by the maps LD and RD of the canonical ωx defined on T ∗ Mx and ωy on T ∗ My : L∗ D ωx = Di,j (x, y)dxi ∧ dyj = −ωD, R∗ Dωy = −Di,j (x, y)dxi ∧ dyj = ωD. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu “Proper” Divergence Functions An almost complex structure J on T(x,y)M × M is a linear isomorphism on the tangent bundle, with J2 = −I. Requiring J to be compatible with ωD, that is, ωD(JX, JY ) = ωD(X, Y ), ∀X, Y ∈ T(x,y)M × M, we may obtain a constraint on the divergence function D Di,j = Dj,i , or explicitly ∂2 D ∂xi ∂yj = ∂2 D ∂xj ∂yi . Note that, by definition, this condition is always satisfied on ∆M. We now require it to be satisfied on M × M (at least a neighborhood U around ∆M). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Proper Divergence Functions (cont) For proper divergence functions, we can induce a Riemannian metric gD on M × M, defined by gD(X, Y ) = ωD(X, JY ). Explicitly, its components are: gij = gD (∂xi , ∂xj ) = −Di,j , g,ij = gD ∂yi , ∂yj = −Dj,i , gi,j = gD ∂xi , ∂yj = 0 . So the Riemannian metric on M × M is gD = −Di,j dxi dxj + dyi dyj . Whereas a divergence function induces a Riemannian structure on the diagonal manifold ∆M of M × M, a proper divergence function induces a Riemannian structure on M × M. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Proper Divergence Functions (cont) For proper divergence functions, we can induce a Riemannian metric gD on M × M, defined by gD(X, Y ) = ωD(X, JY ). Explicitly, its components are: gij = gD (∂xi , ∂xj ) = −Di,j , g,ij = gD ∂yi , ∂yj = −Dj,i , gi,j = gD ∂xi , ∂yj = 0 . So the Riemannian metric on M × M is gD = −Di,j dxi dxj + dyi dyj . Whereas a divergence function induces a Riemannian structure on the diagonal manifold ∆M of M × M, a proper divergence function induces a Riemannian structure on M × M. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 5. Induced K¨ahler Structure on M × M We investigate whether M × M can be complexified (and hence becomes a K¨ahler manifold). That is to say, for (x, y) ∈ M × M, we want z = x + √ −1y. Any manifold equipped with “compatible triple” (symplectic ω, Riemannian g, and almost complex J) has line element ds2 = g − Jω. The line element of M × M induced by a proper D, is ds2 = gD − iωD = −Di,j dzi ⊗ d¯zj where we have used √ −1 in place of J. Now that D(x, y) = D z + ¯z 2 , z − ¯z 2 √ −1 ≡ D(z, ¯z), so ∂2 D ∂zi ∂¯zj = 1 4 (Dij + D,ij ) = 1 2 ∂2 D ∂zi ∂¯zj . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 5. Induced K¨ahler Structure on M × M We investigate whether M × M can be complexified (and hence becomes a K¨ahler manifold). That is to say, for (x, y) ∈ M × M, we want z = x + √ −1y. Any manifold equipped with “compatible triple” (symplectic ω, Riemannian g, and almost complex J) has line element ds2 = g − Jω. The line element of M × M induced by a proper D, is ds2 = gD − iωD = −Di,j dzi ⊗ d¯zj where we have used √ −1 in place of J. Now that D(x, y) = D z + ¯z 2 , z − ¯z 2 √ −1 ≡ D(z, ¯z), so ∂2 D ∂zi ∂¯zj = 1 4 (Dij + D,ij ) = 1 2 ∂2 D ∂zi ∂¯zj . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 5. Induced K¨ahler Structure on M × M We investigate whether M × M can be complexified (and hence becomes a K¨ahler manifold). That is to say, for (x, y) ∈ M × M, we want z = x + √ −1y. Any manifold equipped with “compatible triple” (symplectic ω, Riemannian g, and almost complex J) has line element ds2 = g − Jω. The line element of M × M induced by a proper D, is ds2 = gD − iωD = −Di,j dzi ⊗ d¯zj where we have used √ −1 in place of J. Now that D(x, y) = D z + ¯z 2 , z − ¯z 2 √ −1 ≡ D(z, ¯z), so ∂2 D ∂zi ∂¯zj = 1 4 (Dij + D,ij ) = 1 2 ∂2 D ∂zi ∂¯zj . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu 5. Induced K¨ahler Structure on M × M We investigate whether M × M can be complexified (and hence becomes a K¨ahler manifold). That is to say, for (x, y) ∈ M × M, we want z = x + √ −1y. Any manifold equipped with “compatible triple” (symplectic ω, Riemannian g, and almost complex J) has line element ds2 = g − Jω. The line element of M × M induced by a proper D, is ds2 = gD − iωD = −Di,j dzi ⊗ d¯zj where we have used √ −1 in place of J. Now that D(x, y) = D z + ¯z 2 , z − ¯z 2 √ −1 ≡ D(z, ¯z), so ∂2 D ∂zi ∂¯zj = 1 4 (Dij + D,ij ) = 1 2 ∂2 D ∂zi ∂¯zj . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Condition for M × M to become K¨ahler If, in addition to D being “proper”, we require D to satisfy Dij + D,ij = κDi,j where κ is a constant, then M × M admits a K¨ahler potential ds2 = κ 2 ∂2 D ∂zi ∂¯zj dzi ⊗ d¯zj . This is a sufficient condition for M × M to be K¨ahler. Theorem When M is Hessian, M × M is K¨ahler, with the K¨ahler potential given by 2 1 + α2 Φ(α) (z, ¯z) = 2 1 + α2 Φ 1 − α 2 x + 1 + α 2 y . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Condition for M × M to become K¨ahler If, in addition to D being “proper”, we require D to satisfy Dij + D,ij = κDi,j where κ is a constant, then M × M admits a K¨ahler potential ds2 = κ 2 ∂2 D ∂zi ∂¯zj dzi ⊗ d¯zj . This is a sufficient condition for M × M to be K¨ahler. Theorem When M is Hessian, M × M is K¨ahler, with the K¨ahler potential given by 2 1 + α2 Φ(α) (z, ¯z) = 2 1 + α2 Φ 1 − α 2 x + 1 + α 2 y . Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu Summary: Structures Induced by DΦ-Divergence D (α) Φ (x, y) = 4 1 − α2 1 − α 2 Φ(x) + 1 + α 2 Φ(y)−Φ 1 − α 2 x + 1 + α 2 y Theorem DΦ-divergence induces a family of compatible triples {M, ω(α) , g(α) } on U × U ⊂ M × M such that 1 the symplectic form ω(α) = Φ (α) ij dxi ∧ dyj ; 2 the Riemannian metric g(α) = Φ (α) ij (dxi dxj + dyi dyj ); 3 the K¨ahler structure ds2(α) = Φ (α) ij dzi ⊗ d¯zj = 8 1+α2 ∂2Φ(α) ∂zi ∂¯zj with the K¨ahler potential given by 2 1 + α2 Φ(α) (z, ¯z) where Φ(α) = Φ 1−α 2 x + 1+α 2 y = Φ((1−α 4 + 1+α 4 √ −1 )z + (1−α 4 − 1+α 4 √ −1 )¯z) = Φ(α) (z, ¯z). Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu THANK YOU FOR ATTENTION!! 1 Zhang, J. (2004). Divergence function, duality, and convex analysis. Neural Computation, 16, 159-195. 2 Zhang, J. (2006). Referential duality and representational duality on statistical manifolds. Proceedings of the Second International Symposium on Information Geometry and Its Applications, Tokyo (pp 58-67). 3 Zhang, J. (2007). A note on curvature of α-connections on a statistical manifold. Annals of Institute of Statistical Mathematics, 59, 161-170. 4 Zhang, J. and Matsuzoe, H. (2009). Dualistic differential geometry associated with a convex function. In Gao D.Y. and Sherali, H.D. (Eds) Advances in Applied Mathematics and Global Optimization, Advances in Mechanics and Mathematics, Vol. III, Chapter 13, Springer (pp 439-466). 5 Zhang, J. (under review). Nonparametric information geometry: from divergence function to referential representational biduality on statistical manifolds. Submitted. Jun Zhang University of Michigan, Ann Arbor, MI 48109, USA (junz@umich.edu)Symplectic and K¨ahler Structures on Statistical Manifolds Indu