Our results are obtained in Amari's recently first a review of our theory with special emphasis on the specific issues of Get PDF (1 MB) Cite . Since PEFs have computationally intractable normalization terms, we estimate PEFs with score matching, and consider a projective distance: the symmetrized $$\gamma$$-divergence. BibTex; Full citation Abstract. In three of the cases we consider, we illustrate a We provide an analysis on a list of commonly used components, and demonstrate how to use this concept to further improve optimization. Bregman divergences form a rich and versatile family of distances that unifies quadratic Euclidean distances with various well-known statistical entropic measures. We also provide a complete differential-geometric characterization of the primal-dual curvature in the primal-dual algorithm. metrizable topological spaces provided with Borel measures. MULTIPARAMETER CURVED EXPONENTIAL FAMILIES. line of research with respect to other approaches is briefly discussed. The introduction of a metric onto the space of parameters in models in statistical mechanics and beyond gives an alternative perspective on their phase structure. Assuming that the users have the channels’ statistical information and adopt the minimum mean square error (MMSE) receivers, this article studies how to design an optimal training sequence to minimize the total weighted mean square error (MSE) of channel estimation. In this paper a brief relatively nontechnical account is given of some relevant ideas in differential geometry. It is then applied to Bayesian inference by considering the Jeffreys prior determined by the volume form. The advantage of this MCIG is that it allows a practical use of the Bregman algorithmic toolbox on a wide range of probability distribution families. In the present paper Čencov's result is extended to the positive cone. To read the file of this research, you can request a copy directly from the author. We show that Bhattacharyya distances on members of the same statistical exponential family amount to calculate a Burbea-Rao divergence in disguise. Finally, we show how to compute statistical H\"older centroids with respect to those divergences, and carry out center-based clustering toy experiments on a set of Gaussian distributions that demonstrate empirically that symmetrized H\"older divergences outperform the symmetric Cauchy-Schwarz divergence. Four problems related to information divergence measures defined on finite The information geometry induced by the Kullback-Leibler (KL) divergence yields a dually flat space where the KL divergence between two w-mixtures amounts to a Breg-man divergence for the negative Shannon entropy generator , called the Shannon information. Given a manifold $$\mathfrak {M}$$, a divergence function$$\mathcal {D}$$ is a smooth, nonnegative function on the product manifold $$\mathfrak {M}\times \mathfrak {M}$$ that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold $$\varDelta _{\mathfrak {M}} \subset \mathfrak {M}\times \mathfrak {M}$$. It is then shown that the class of $$\mathcal {D}_\varPhi$$-divergence functions [23], as induced by a strictly convex function$$\varPhi$$ on $$\mathfrak {M}$$, satisfies all these requirements and hence makes $$\mathfrak {M}\times \mathfrak {M}$$ a Kähler manifold (with Kähler potential given by $$\varPhi$$). The new definition of a canonical divergence reduces to the known canonical divergence in the case of dual flatness. By considering the geometry of the underlying statistical manifold, we define exactly the solution of the optimization problem as the unique intersection of a geodesic with a dual hyperplane. We then proceed by defining skew Burbea-Rao divergences, and show that skew Burbea-Rao divergences amount in limit cases to compute Bregman divergences. In fact, we will show here that, if f : ℝn → ℝ is convex and x ∈ ℝn is fixed, then the function  The performance of our method was evaluated by numerical examples and manipulator experiments.

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