(2015) GSI2015

The information geometry of mirror descent Garvesh Raskutti, Sayan Mukherjee GSI2015
Détails de l'article
We prove the equivalence of two online learning algorithms, mirror descent and natural gradient descent. Both mirror descent and natural gradient descent are generalizations of online gradient descent when the parameter of interest lies on a non-Euclidean manifold. Natural gradient descent selects the steepest descent direction along a Riemannian manifold by multiplying the standard gradient by the inverse of the metric tensor. Mirror descent induces non-Euclidean structure by solving iterative optimization problems using different proximity functions. In this paper, we prove that mirror descent induced by a Bregman divergence proximity functions is equivalent to the natural gradient descent algorithm on the Riemannian manifold in the dual coordinate system.We use techniques from convex analysis and connections between Riemannian manifolds, Bregman divergences and convexity to prove this result. This equivalence between natural gradient descent and mirror descent, implies that (1) mirror descent is the steepest descent direction along the Riemannian manifold corresponding to the choice of Bregman divergence and (2) mirror descent with log-likelihood loss applied to parameter estimation in exponential families asymptotically achieves the classical Cramér-Rao lower bound.
The information geometry of mirror descent
Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic Stastistical Space Form Hasan Shahid, Jamali Mohammed, Michel Nguiffo Boyom GSI2015
Détails de l'article
In this article, we derive an inequality satisfied by the squared norm of the imbedding curvature tensor of Multiply CR-warped product statistical submanifolds N of holomorphic statistical space forms M. Furthermore, we prove that under certain geometric conditions, N and M become Einstein.
Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic Stastistical Space Form
Path connectedness on a space of probability density functions Osamu Komori, Shinto Eguchi GSI2015
Détails de l'article
We introduce a class of paths or one-parameter models connecting arbitrary two probability density functions (pdf’s). The class is derived by employing the Kolmogorov-Nagumo average between the two pdf’s. There is a variety of such path connectedness on the space of pdf’s since the Kolmogorov-Nagumo average is applicable for any convex and strictly increasing function. The information geometric insight is provided for understanding probabilistic properties for statistical methods associated with the path connectedness. The one-parameter model is extended to a multidimensional model, on which the statistical inference is characterized by sufficient statistics.
Path connectedness on a space of probability density functions
Computational Information Geometry: mixture modelling Frank Critchley, Germain Van Bever, Paul Marriott, Radka Sabolova GSI2015
Geometry on the set of quantum states and quantum correlations Dominique Spehner GSI2015
Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems Charles-Michel Marle GSI2015
Détails de l'article
I will present some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. Lie group and Lie algebra actions on symplectic and Poisson manifolds, momentum maps and their equivariance properties, first integrals associated to symmetries of Hamiltonian systems will be discussed. Reduction methods taking advantage of symmetries will be discussed.
Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems

(2015) ETTC 2015

Proceedings ETTC 2015.zip ETTC 2015
Détails de l'article
This zip file contains all ETTC 2015 communications and the final programme
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