Information Geometry and its Applications III

Abstract Fumio Hiai

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Fumio Hiai  (Tohoku University, Japan)
Wednesday, August 04, 2010, room Hörsaal 1
Riemannian metrics on positive definite matrices related to means

This is a joint work with D. Petz and a continuation of the paper in LAA 430 (2009) with the same title. The formula39 Hermitian matrices form the formula41-dimensional Euclidean space with respect to Hilbert-Schmidt inner product. The set formula43 of all formula39 positive definite matrices, being an open subset of formula47, is naturally equipped with a formula49 manifold structure. An smooth kernel function formula51 induces a Riemannian metric formula53 on formula43 defined by
displaymath57
where formula59 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 formula61 of the form formula63, a formula65 (formula67)-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 formula43 of this type.

(1) Since the Riemannian manifold formula75 with formula77 is complete if and only if formula79, the existence of geodesic shortest curves in the case formula81 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 formula65 but independent of the choice of M. Moreover, we show the existence of a geodesic shortest curve joining formula91 for the metric formula53 with formula77 if formula65 is sufficiently near 2.

(2) We present a necessary and sufficient condition for Riemannian metrics formula53 and formula103 induced by formula77 and formula107 to be isometric under the transformation formula109 given by a smooth function formula111. The condition is explicitly given in terms of formula113, and formula115.

(3) From the above (2) we can construct a one-parameter isometric family of Riemannian metrics starting from any formula103 inside the set of Riemannian metrics we are treating. Those isometric families have different features between the cases formula119 and formula121. We see that each of those families converges to the metric formula123 induced by the square of the logarithmic mean formula125. Thus formula123 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 formula123 is formula131 (formula133), this shows the Riemannian geometric interpretation for the limit formulas such as
align23
where formula135 is the t-power mean (formula133).

 

     

Date and Location

August 02 - 06, 2010
University of Leipzig
Augustusplatz
04103 Leipzig
Germany

Scientific Organizers

Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Information Theory of Cognitive Systems Group
Germany
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Paolo Gibilisco
Università degli Studi di Roma "Tor Vergata"
Facoltà di Economia
Italy
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František Matúš
Academy of Sciences of the Czech Republic
Institute of Information Theory and Automation
Czech Republic
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Scientific Committee

Shun-ichi Amari
RIKEN
Brain Science Institute, Mathematical Neuroscience Laboratory
Japan
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Imre Csiszár
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
Hungary
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Dénes Petz
Budapest University of Technology and Economics
Department for Mathematical Analysis
Hungary
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Giovanni Pistone
Collegio Carlo Alberto, Moncalieri
Italy
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Administrative Contact

Antje Vandenberg
Max Planck Institute for Mathematics in the Sciences
Contact by Email
Phone: (++49)-(0)341-9959-552
Fax: (++49)-(0)341-9959-555

05.04.2017, 12:42