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Riemannian metrics on positive definite matrices related to means

  • Fumio Hiai (Tohoku University, Japan)
Raum n.n. Universität Leipzig (Leipzig)

Abstract

This is a joint work with D.\ Petz and a continuation of the paper in LAA {\bf 430} (2009) with the same title. The $n\times n$ Hermitian matrices form the $n^2$-dimensional Euclidean space with respect to Hilbert-Schmidt inner product. The set $\mathbb{P}_n$ of all $n\times n$ positive definite matrices, being an open subset of $\mathbb{H}_n$, is naturally equipped with a $C^\infty$ manifold structure. An smooth kernel function $\phi:(0,\infty)\times(0,\infty)\to(0,\infty)$ induces a Riemannian metric $K^\phi$ on $\mathbb{P}_n$ defined by $$ K_D^\phi(H,K):=\sum_{i,j}\phi(\lambda_i,\lambda_j)^{-1}\mathrm{Tr}\, P_iHP_jK, \qquad D\in\mathbb{P}_n,\ H,K\in\mathbb{H}_n, $$ where $D=\sum_i\lambda_iP_i$ is the spectral decomposition. Certain mportant quantities in quantum information geometry, such as statistical metric, quantum Fisher informations, and quantum variances, are Riemannian metrics arising from kernel functions $\phi$ of the form $M(x,y)^\theta$, a $\theta$ ($\in\mathbb{R}$)-power of a symmetric homogenous mean $M(x,y)$ of $x,y>0$. We discuss the following topics concerning geodesic shortest curves and geodesic distance of Riemannian metrics on $\mathbb{P}_n$ of this type.

(1) Since the Riemannian manifold $(\mathbb{P}_n,K^\phi)$ with $\phi=M(x,y)^\theta$ is complete if and only if $\theta=2$, the existence of geodesic shortest curves in the case $\theta\ne2$ does not seem obvious. When $A,B$ are commuting, we present an explicit formula of a geodesic shortest curve between $A,B$ that is depending on $\theta$ but independent of the choice of $M$. Moreover, we show the existence of a geodesic shortest curve joining $A,B\in\mathbb{P}_n$ for the metric $K^\phi$ with $\phi=M(x,y)^\theta$ if $\theta$ is sufficiently near $2$.

(2) We present a necessary and sufficient condition for Riemannian metrics $K^\phi$ and $K^\psi$ induced by $\phi=M(x,y)^\theta$ and $\psi=N(x,y)^\kappa$ to be isometric under the transformation $D\in\mathbb{P}_n\mapsto F(D)\in\mathbb{P}_n$ given by a smooth function $F:(0,\infty)\to(0,\infty)$. The condition is explicitly given in terms of $M,N,\theta$, and $\kappa$.

(3) From the above (2) we can construct a one-parameter isometric family of Riemannian metrics starting from any $K^\psi$ inside the set of Riemannian metrics we are treating. Those isometric families have different features between the cases $\kappa\ne2$ and $\kappa=2$. We see that each of those families converges to the metric $K^{M_{\mathrm{L}}^2}$ induced by the square of the logarithmic mean $M_{\mathrm{L}}(x,y):=(x-y)/(\log x-\log y)$. Thus $K^{M_{\mathrm{L}}^2}$ may be regarded as a (unique) attractor in the set of all Riemannian metrics of our discussion. From the fact that the geodesic shortest curve for the metric $K^{M_{\mathrm{L}}^2}$ is $\gamma(t)=\exp((1-t)\log A+\log B)$ ($0\le t\le1$), this shows the Riemannian geometric interpretation for the limit formulas such as \begin{align*} \lim_{\alpha\to0}((1-t)A^\alpha+tB^\alpha)^{1/\alpha}&=\exp((1-t)\log A+\log B), \\ \lim_{\alpha\to0}(A^\alpha\,\#_t\,B^\alpha)^{1/\alpha}&=\exp((1-t)\log A+\log B), \end{align*} where $\#_t$ is the $t$-power mean ($0\le t\le1$).

conference
8/2/10 8/6/10

Information Geometry and its Applications III

Universität Leipzig Raum n.n.

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Nihat Ay

Max Planck Institute for Mathematics in the Sciences, Germany

Paolo Gibilisco

Università degli Studi di Roma "Tor Vergata", Italy

František Matúš

Academy of Sciences of the Czech Republic, Czech Republic