Gaussian distributions on the space of symmetric positive definite matrices from Souriau’s Gibbs state for Siegel domains by coadjoint orbit and moment map. (English) Zbl 1486.60024
Nielsen, Frank (ed.) et al., Geometric science of information. 5th international conference, GSI 2021, Paris, France, July 21–23, 2021. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 12829, 245-255 (2021).
Summary: We will introduce Gaussian distribution on the space of Symmetric Positive Definite (SPD) matrices, through Souriau’s covariant Gibbs density by considering this space as the pure imaginary axis of the homogeneous Siegel upper half space where Sp (2n,R)/U(n) acts transitively. Gauss density of SPD matrices is computed through Souriau’s moment map and coadjoint orbits. We will illustrate the model first for Poincaré unit disk, then Siegel unit disk and finally upper half space. For this example, we deduce Gauss density for SPD matrices.
For the entire collection see [Zbl 1482.94007].
For the entire collection see [Zbl 1482.94007].
MSC:
| 60E05 | Probability distributions: general theory |
| 62B11 | Information geometry (statistical aspects) |
| 15A63 | Quadratic and bilinear forms, inner products |
| 82B30 | Statistical thermodynamics |
Keywords:
symmetric positive definite matrices; Lie groups thermodynamics; symplectic geometry; maximum entropy; exponential density familyReferences:
| [1] | Barbaresco, F., Lie group statistics and lie group machine learning based on Souriau lie groups Thermodynamics & Koszul-Souriau-Fisher metric: new entropy definition as generalized Casimir invariant function in Coadjoint representation, Entropy, 22, 642 (2020) · doi:10.3390/e22060642 |
| [2] | Barbaresco, F.; Gay-Balmaz, F., Lie group cohomology and (multi)symplectic integrators: new geometric tools for lie group machine learning based on Souriau geometric statistical mechanics, Entropy, 22, 498 (2020) · doi:10.3390/e22050498 |
| [3] | Chevallier, E.; Forget, T.; Barbaresco, F.; Angulo, J., Kernel density estimation on the Siegel space with an application to radar processing, Entropy, 18, 396 (2016) · doi:10.3390/e18110396 |
| [4] | Cishahayo, C.; de Bièvre, S., On the contraction of the discrete series of SU(1;1), Annales de l’institut Fourier, tome, 43, 2, 551-567 (1993) · Zbl 0793.22005 · doi:10.5802/aif.1346 |
| [5] | Renaud, J., The contraction of the SU(1, 1) discrete series of representations by means of coherent states, J. Math. Phys., 37, 7, 3168-3179 (1996) · Zbl 0860.22013 · doi:10.1063/1.531563 |
| [6] | Cahen, B.: Contraction de SU(1,1) vers le groupe de Heisenberg, Travaux de Mathématiques. Fascicule XV, 19-43 (2004) · Zbl 1074.22005 |
| [7] | Cahen, B.: Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group, Acta. Univ. Palacki. Olomuc. Fac. rer. Nat., Mat. 52, n°&, pp.35-48, 2013 · Zbl 1296.22007 |
| [8] | Berezin, FA, Quantization in complex symmetric space, Math, USSR Izv, 9, 341-379 (1975) · Zbl 0312.53050 · doi:10.1070/IM1975v009n02ABEH001480 |
| [9] | Marle, C-M, On gibbs states of mechanical systems with symmetries, JGSP, 57, 45-85 (2020) · Zbl 1459.53074 · doi:10.7546/jgsp-57-2020-45-85 |
| [10] | Marle, C.-M.: On Generalized Gibbs states of mechanical systems with symmetries, arXiv:2012.00582v2 [math.DG] (2021) |
| [11] | Marle, C.-M. Projection Stereographique et moments, Hal-02157930, Version 1; June 2019. https://hal.archives-ouvertes.fr/hal-02157930/. Accessed 31 May 2020 |
| [12] | Hua, L.K.: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Translations of Mathematical Monographs, vol. 6, American Mathematical Society, Providence (1963) · Zbl 0112.07402 |
| [13] | Nielsen, F., The Siegel-Klein disk, Entropy, 22, 1019 (2020) · doi:10.3390/e22091019 |
| [14] | Ohsawa, T., Tronci, C.: Geometry and dynamics of Gaussian wave packets and their Wigner transforms. J. Math. Phys. 58, 092105 (2017) · Zbl 1372.81104 |
| [15] | Siegel, CL, Symplectic geometry, Am. J. Math., 65, 1, 1-86 (1943) · Zbl 0063.07003 · doi:10.2307/2371774 |
| [16] | Leverrier, A., SU(p, q) coherent states and a Gaussian de Finetti theorem, J. Math. Phys., 59, 042202 (2018) · Zbl 1401.81056 · doi:10.1063/1.5007334 |
| [17] | Satake I.: Algebraic Structures of Symmetric Domains, Princeton University Press, Princeton (1980) · Zbl 0483.32017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
