Geometry on Positive Definite Matrices Induced from V-potential functions

28/08/2013
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Geometry on Positive Definite Matrices Induced from V-potential functions

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Geometry on Positive Definite Matrices Induced from V-potential functions Atsumi Ohara University of Fukui & Shinto Eguchi Institute of Statistical Mathematics Geometric Science of Information 2013 at Ecole des Mines, 2013 August 29 1 1. Introduction  : the set of Positive Definite real symmetric matrices ● logarithmic characteristic func. on [Vinberg 63], [Faraut & Koranyi 94] R);(,detlog)( nPDPPP Î-=j 2 2 φ(P) = -log det P appears in  Semidefinite Programming (SDP) self-concordant barrier function  Multivariate Analysis (Gaussian dist.) log-likelihood function (structured covariance matrix estimation)  Symmetric cone: log characteristic function  Information geometry on a potential function in standard case 3 Information geometry on Dualistic geometrical structure : arbitrary vector fields on :Riemannian metric , :a pair of dual affine connections 4 standard IG on [O,Suda & Amari LAA96] - : Riemannian metric is the Hesse matrix of (Fisher for Gaussian))(Pj - , :related to the third derivatives of )(Pj Nice properties: -invariant (unique), KL-divergence, Pythagorean theorem, etc plays a role of potential function)(Pj 5 Our interests in applying the standard IG on  Structures of stable matrices and stabilizing FB gains [O & Amari, Kybernetika93]  dual conections and Jordan algebra [Uohashi & O, Positivity04]  means on symmetric cones [O, IEOT04]  complexity analysis of IPM on sym. cones [Kakihara, O & Tsuchiya, Friday] 6 Purpose of this presentation  The other convex potentials V-potential functions  Their different and/or common geometric structures 7 Outline  V-potential function  Dualistic geometry on  Foliated Structure  Decomposition of divergence  Applications  geometry of a family of multivariate elliptic distributions  new update formulas for the quasi-Newton algorithm [Kanamori & O, OMS13] 8 9 2. Preliminaries and Notation  the set of n by n real symmetric matrix  :arbitrary set of basis matrices  (primal) affine coordinate system  Identification 3. V-potential function -The standard case: PPssV detlog)(log)( -=Þ-= j , RR ®+:)(sV 10 Def. Def. Rem. The standard case V= -log: 2,0)(,1)(1 ³=-= kss knn Prop.1 (convexity condition) The Hessian matrix of the V-potential is positive definite on if and only if 11 Prop. When two conditions in Prop.1 hold, Riemannian metric derived from the V-potential is = Here, X, Y in sym(n,R) ~ tangent vectors at P Rem. The standard case V= -log: = 12 Prop. (affine connections) Let be the canonical flat connection on . Then the V-potential defines the following dual connection with respect to : Ñ 13 Rem. the standard case V= - log: “mutation” of the Jordan product of Ei and Ej [Uohashi & O, Positivity04] 14 divergence function Divergence derived from : - a variant of relative entropy, - Pythagorean type decomposition 15 15 4. Invariance of the structure on  Linear transformation on congruent transformation: the differential: T G T G GXGX nGLGGPGP = Î= *)( ),,(, t t R 16 16 4. Invariance of the structure on  Linear transformation on congruent transformation: the differential:  Invariance  metric:  connections: and the same for where T G T G GXGX nGLGGPGP = Î= *)( ),,(, t t R 17 17 Prop. The largest group that preserves the dualistic structure invariant is except in the standard case. ),( RnSLGG Ît ),( RnGLGG ÎtRem. the standard case: Rem. The power potential of the form: has a special property. 18 with with Special properties for the power potentials 1) Orthogonality is GL(n)-invariant. 2) The dual affine connections derived from the power potentials are GL(n)-invariant. Hence,  Both - and -projections are GL(n) - invariant. 19 5. Foliated Structures The following foliated structure features the dualistic geometry derived by the V-potential. 20 leaf ray Prop. Each leaf and ray are orthogonal each other with respect to . Prop. Every is simultaneously a - and - geodesic for an arbitrary V-potential. 21 Prop. Each leaf is a homogeneous space with the constant negative curvature 22 6. Decomposition of divergence  Each is a level surface of both and Prop. If and with , then where 23 24  Each is of constant curvature ks By combining the modified Pythagorean thm [Kurose 94] If and are orthogonal at R, then Prop. Application to multivariate statistics  Non Gaussian distribution (generalized exponential family)  Robust statistics  beta-divergence,  Machine learning, and so on  Nonextensive statistical physics  Power distribution,  generalized (Tsallis) entropy, and so on 25 Application to multivariate statistics: Geometry of U-model  U-model Def. Given a convex function U on R and set u=U’, U-model is a family of elliptic probability distributions specified by P: :normalizing const. 26 Rem. When U=exp, the U-model is the family of Gaussian distributions. U-divergence: Natural closeness measure on the U-model Rem. When U=exp, the U-divergence is the Kullback-Leibler divergence (relative entropy). 27 Prop. Geometry of the U-model equipped with the U-divergence coincides with derived from the following V-potential function: 28 Conclusions  Derived dualistic geometry is invariant under the SL(n,R)-group actions  For power funcion, dual connections and orthogonality are GL(n,R)-invariant  Each leaf is a homogeneous manifold with a negative constant curvature  Decomposition of the divergence function  Relation with the U-model with the U-divergence 29 Main References A. Ohara, N. Suda and S. Amari, Dualistic Differential Geometry of Positive Definite Matrices and Its Applications to Related Problems, Linear Algebra and its Applications, Vol.247, 31-53 (1996). A. Ohara, Geodesics for Dual Connections and Means on Symmetric Cones, Integral Equations and Operator Theory, Vol.50, 537-548 (2004). A. Ohara and S. Eguchi, Geometry on positive definite matrices and V-potential function, Research Memorandum No. 950, The Institute of Statistical Mathematics, Tokyo, July (2005). T. Kanamori and A. Ohara, A Bregman Extension of quasi-Newton updates I: An Information Geometrical Framework, Optimization Methods and Software, Vol. 28, No. 1, 96-123 (2013). 30