Group Theoretical Study on Geodesics for the Elliptical Models

28/10/2015
Auteurs : Hiroto Inoue
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14266

Résumé

We consider the geodesic equation on the elliptical model, which is a generalization of the normal model. More precisely, we characterize this manifold from the group theoretical view point and formulate Eriksen’s procedure to obtain geodesics on normal model and give an alternative proof for it.

Group Theoretical Study on Geodesics for the Elliptical Models

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We consider the geodesic equation on the elliptical model, which is a generalization of the normal model. More precisely, we characterize this manifold from the group theoretical view point and formulate Eriksen’s procedure to obtain geodesics on normal model and give an alternative proof for it.

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Group Theoretical Study on Geodesics for the Elliptical Models Hiroto Inoue Kyushu University, Japan October 28, 2015 GSI2015, ´Ecole Polytechnique, Paris-Saclay, France Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 1 / 14 Overview 1 Eriksen’s construction of geodesics on normal model Problem 2 Reconsideration of Eriksen’s argument Embedding Nn → Sym+ n+1(R) 3 Geodesic equation on Elliptical model 4 Future work Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 2 / 14 Eriksen’s construction of geodesics on normal model Let Sym+ n (R) be the set of n-dimensional positive-definite matrices. The normal model Nn = (M, ds2) is a Riemannian manifold defined by M = (µ, Σ) ∈ Rn × Sym+ n (R) , ds2 = (t dµ)Σ−1 (dµ) + 1 2 tr((Σ−1 dΣ)2 ). The geodesic equation on Nn is ¨µ − ˙ΣΣ−1 ˙µ = 0, ¨Σ + ˙µt ˙µ − ˙ΣΣ−1 ˙Σ = 0. (1) The solution of this geodesic equation has been obtained by Eriksen. Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 3 / 14 Theorem ([Eriksen 1987]) For any x ∈ Rn, B ∈ Symn(R), define a matrix exponential Λ(t) by Λ(t) =   ∆ δ Φ tδ tγ tΦ γ Γ   := exp(−tA), A :=   B x 0 tx 0 −tx 0 −x −B   ∈ Mat2n+1. (2) Then, the curve (µ(t), Σ(t)) := (−∆−1δ, ∆−1) is the geodesic on Nn satisfiying the initial condition (µ(0), Σ(0)) = (0, In), ( ˙µ(0), ˙Σ(0)) = (x, B). (proof) We see that by the definition, (µ(t), Σ(t)) satisfies the geodesic equation. Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 4 / 14 Problem 1 Explain Eriksen’s theorem, to clarify the relation between the normal model and symmetric spaces. 2 Extend Eriksen’s theorem to the elliptical model. Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 5 / 14 Reconsideration of Eriksen’s argument Sym+ n+1(R) Notice that the positive-definite symmetric matrices Sym+ n+1(R) is a symmetric space by G/K Sym+ n+1(R) gK → g · tg, where G = GLn+1(R), K = O(n + 1). This space G/K has the G-invariant Riemannian metric ds2 = 1 2 tr (S−1 dS)2 . Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 6 / 14 Embedding Nn → Sym+ n+1(R) Put an affine subgroup GA := P µ 0 1 P ∈ GLn(R), µ ∈ Rn ⊂ GLn+1(R). Define a Riemannian submanifold as the orbit GA · In+1 = {g · t g| g ∈ GA} ⊂ Sym+ n+1(R). Theorem (Ref. [Calvo, Oller 2001]) We have the following isometry Nn ∼ −→ GA · In+1 ⊂ Sym+ n+1(R), (Σ, µ) → Σ + µtµ µ tµ 1 . (3) Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 7 / 14 Embedding Nn → Sym+ n+1(R) By using the above embedding, we get a simpler expression of the metric and the geodesic equation. Nn ∼= GA · In+1 ⊂ Sym+ n+1(R) coordinate (Σ, µ) → S = Σ + µtµ µ tµ 1 metric ds2 = (tdµ)Σ−1(dµ) +1 2tr((Σ−1dΣ)2) ⇔ ds2 = 1 2 tr (S−1dS)2 geodesic eq. ¨µ − ˙ΣΣ−1 ˙µ = 0, ¨Σ + ˙µt ˙µ − ˙ΣΣ−1 ˙Σ = 0 ⇔ (In, 0)(S−1 ˙S) = (B, x) Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 8 / 14 Reconsideration of Eriksen’s argument We can interpret the Eriksen’s argument as follows. Differential equation Geodesic equation Λ−1 ˙Λ = −A −→ (In, 0)(S−1 ˙S) = (B, x) A =   B x 0 t x 0 −t x 0 −x −B   −→ e−tA =   ∆ δ ∗ t δ ∗ ∗ ∗ ∗   −→ S := ∆ δ t δ −1 ∈ ∈ ∈ {A : JAJ = −A} −→ {Λ : JΛJ = Λ−1 } −→ Essential! Nn ∼= GA · In+1 ∩ ∩ ∩ sym2n+1(R) −→ exp Sym+ 2n+1(R) −→ projection Sym+ n+1(R) Here J =   In 1 In  . Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 9 / 14 Geodesic equation on Elliptical model Definition Let us define a Riemannian manifold En(α) = (M, ds2) by M = (µ, Σ) ∈ Rn × Sym+ n (R) , ds2 = (t dµ)Σ−1 (dµ) + 1 2 tr((Σ−1 dΣ)2 )+ 1 2 dα tr(Σ−1 dΣ) 2 . (4) where dα = (n + 1)α2 + 2α, α ∈ C. Then En(0) = Nn. The geodesic equation on En(α) is    ¨µ − ˙ΣΣ−1 ˙µ = 0, ¨Σ + ˙µt ˙µ − ˙ΣΣ−1 ˙Σ− dα ndα + 1 t ˙µΣ−1 ˙µΣ = 0. (5) This is equivalent to the geodesic equation on the elliptical model. Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 10 / 14 Geodesic equation on Elliptical model The manifold En(α) is also embedded into positive-definite symmetric matrices Sym+ n+1(R), ref. [Calvo, Oller 2001], and we have simpler expression of the geodesic equation. En(α) ∼= ∃GA(α) · In+1 ⊂ Sym+ n+1(R) coordinate (Σ, µ) → S = |Σ|α Σ + µtµ µ tµ 1 metric (4) ⇔ ds2 = 1 2 tr (S−1dS)2 geodesic eq. (5) ⇔ (In, 0)(S−1 ˙S) = (C, x) − α(log |S|) (In, 0) |A| = det A Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 11 / 14 Geodesic equation on Elliptical model But, in general, we do not ever construct any submanifold N ⊂ Sym+ 2n+1(R) such that its projection is En(α): Differential equation Geodesic equation Λ−1 ˙Λ = −A −→ (In, 0)(S−1 ˙S) = (C, x) − α(log |S|) (In, 0) Λ(t) −→ S(t) ∈ ∈ N −→ En(α) ∼= GA(α) · In+1 ∩ ∩ Sym+ 2n+1(R) −→ projection Sym+ n+1(R) The geodesic equation on elliptical model has not been solved. Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 12 / 14 Future work 1 Extend Eriksen’s theorem for elliptical models (ongoing) 2 Find Eriksen type theorem for general symmetric spaces G/K Sketch of the problem: For a projection p : G/K → G/K, find a geodesic submanifold N ⊂ G/K, such that p|N maps all the geodesics to the geodesics: ∀Λ(t): Geodesic −→ p(Λ(t)): Geodesic ∈ ∈ N −→ p|N p(N) ∩ ∩ G/K −→ p:projection G/K Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 13 / 14 References Calvo, M., Oller, J.M. A distance between elliptical distributions based in an embedding into the Siegel group, J. Comput. Appl. Math. 145, 319–334 (2002). Eriksen, P.S. Geodesics connected with the Fisher metric on the multivariate normal manifold, pp. 225–229. Proceedings of the GST Workshop, Lancaster (1987). Hiroto Inoue (Kyushu Uni.) Group Theoretical Study on Geodesics October 28, 2015 14 / 14