Dually flat manifold
WebDec 8, 2024 · Consider the following partial differential equation for a function $\psi:S\to\mathbb {R}$ : $$\partial_ {i}\psi=\eta_ {i}$$ we may rewrite this as $d\psi=\eta_ … WebOct 1, 2014 · Dually flat Finsler metrics arise from α-flat information structures on Riemann-Finsler manifolds. Such Finsler metrics was introduced by Amari-Nagaoka and Z. Shen (See [3, 72]).
Dually flat manifold
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WebOwens and Go Industries mud flaps attach to your truck using a bracket kit. Each mud flap set includes brackets. Molded models like Husky and Luverne attach using mounting hardware and existing holes. Additional … WebMay 8, 2024 · It is known that such a manifold is dually projectively flat with constant negative sectional curvature, and is closely related to the $\mathcal{F}^{(\alpha)}$-family, a generalized exponential family introduced by the second author. Our main result constructs a dual foliation of the statistical manifold, i.e., an orthogonal decomposition ...
WebFor example, a dually flat geometry can be built on homogeneous cones with the characteristic function F of the cone [48]. Figure 8 illustrates several common constructions of dually flat spaces ... However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as … See more In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; … See more Let $${\displaystyle F:\Omega \to \mathbb {R} }$$ be a continuously-differentiable, strictly convex function defined on a convex set $${\displaystyle \Omega }$$. The Bregman distance associated with F for points $${\displaystyle p,q\in \Omega }$$ is … See more Bregman divergences can be interpreted as limit cases of skewed Jensen divergences (see Nielsen and Boltz, 2011). Jensen … See more In machine learning, Bregman divergences are used to calculate the bi-tempered logistic loss, performing better than the softmax function with noisy datasets. See more • Non-negativity: $${\displaystyle D_{F}(p,q)\geq 0}$$ for all $${\displaystyle p}$$, $${\displaystyle q}$$. This is a consequence of the convexity of $${\displaystyle F}$$ See more A key tool in computational geometry is the idea of projective duality, which maps points to hyperplanes and vice versa, while preserving incidence and above-below relationships. … See more Bregman divergences can also be defined between matrices, between functions, and between measures (distributions). Bregman … See more
WebApr 30, 2024 · Right in the first chapter, the author talks about parallel transport in Dually flat manifolds. Just some quick notation: In the first chapter, the author introduces the … WebMar 16, 2009 · ABSTRACT. Information geometry emerged from studies on invariant properties of a manifold of probability distributions. It includes convex analysis and its …
WebApr 14, 2014 · Information geometry studies the dually flat structure of a manifold, highlighted by the generalized Pythagorean theorem. The present paper studies a class of Bregman divergences called the (ρ,τ)-divergence. A (ρ,τ) -divergence generates a dually flat structure in the manifold of positive measures, as well as in the manifold of positive …
WebIn this section, we shall study the dually flat Minkowskian product Finsler metrics. We need the following lemma. Lemma 3.1 [14] Let F be a Finsler metric on a manifold M, then F is said to be dually flat if and only if at any point there is an adapted local coordinate system (x α) on M such that it satisfies the following equations (3.1) G β ... jeremy darty facebookWebFeb 2, 2016 · A dually flat manifold is a generalization of a Euclidean space, inheriting useful properties from it. A general non-flat manifold is regarded as a curved submanifold of a … pacific rim hayleyWebAug 2, 2024 · For such a manifold, we give a geometric interpretation of its sectional curvature in terms of how the divergence between a pair of primal and dual geodesics differ from the dually flat case. Further results can be found in our follow-up paper [ 27 ] which uncovers a novel relation between optimal transport and information geometry. pacific rim hangarWebWe provide an Information-Geometric formulation of accelerated natural gradient on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. jeremy dauber columbia universityWebDec 8, 2024 · Simple question on parallel transport in dually flat manifolds. 0. Riemannian connection is flat iff there exists a Euclidean coordinate system. 1. Locality of canonical Bregman divergence on dually flat manifolds. 1. Question about connection coefficient on … jeremy darroch companies houseWebJul 1, 2024 · In the case where h is non-degenerate everywhere, it coincides with a dually flat (Hessian) manifold, i.e., T M = E = E and ∇ E and ∇ E are mutually dual flat connections on T M . ... jeremy dale bardstown high schoolWebJun 1, 2002 · Dually flat manifolds constitute fundamental mathematical objects of information geometry. This note establishes several results on the global properties and … pacific rim hitachi