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Dually flat manifold

WebDec 1, 2024 · For a Riemannian manifold (M, g, ∇) prescribed with a flat connection ∇ to become a Hessian manifold with dually flat connections, ∇ must be Codazzi-coupled to g. So for (∇, g) to deviate from a dually flat structure, we either (a) allow ∇ to carry curvature; or (b) no longer require ∇ to be Codazzi-coupled to g. WebJun 1, 2002 · Dually flat manifolds constitute fundamental mathematical objects of information geometry. This note establishes several results on the global properties and …

Dually flat and projectively flat Minkowskian product

WebInformation geometry emerged from studies on invariant properties of a manifold of probability distributions. It includes convex analysis and its duality as a special but important part. Here, we begin with a convex function, and construct a dually flat manifold. The manifold possesses a Riemannian metric, two types of geodesics, and a ... WebApr 13, 2024 · A model of spacetime is presented. It has an extension to five dimensions, and in five dimensions the geometry is the dual of the Euclidean geometry w.r.t. an arbitrary positive-definite metric. Dually flat geometries are well-known in the context of information geometry. The present work explores their role in describing the geometry of spacetime. … reared definition psychology https://bioanalyticalsolutions.net

On some dually flat Finsler metrics with orthogonal invariance

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 WebDec 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 … WebMar 16, 2009 · This work begins with a convex function, and construction of a dually flat manifold, which possesses a Riemannian metric, two types of geodesics, and a … rear edge meaning

Manifold, Divergence and Dually Flat Structure

Category:Characterization of Locally Dually Flat Special Finsler α,β

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Dually flat manifold

Information Geometry and Its Applications: Convex …

WebAug 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. WebSep 29, 2024 · In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences. The exposition is self-contained by concisely introduci …

Dually flat manifold

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WebThe latter class includes the Bregman divergence , also called “geometric divergence” , which turns out to be identical to the “canonical divergence” on a dually flat manifold expressed in a pair of biorthogonal coordinates; those coordinates are induced by a pair of conjugate convex functions via the Legendre–Fenchel transform [2,20]. WebAbout Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators ...

WebMar 20, 2024 · Exponential families and mixture families are parametric probability models that can be geometrically studied as smooth statistical manifolds with respect to any statistical divergence like the Kullback-Leibler (KL) divergence or the Hellinger divergence. When equipping a statistical manifold with the KL divergence, the induced 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_ …

WebFeb 2, 2016 · A dually flat manifold admits two types of foliations, e-foliation and m-foliation, which are orthogonal to each other. This structure is useful for separating two quantities, one represented in the e-coordinates and the other in the m-coordinates. This fits particularly well for analyzing a hierarchical system (Amari 2001). 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]).

WebJan 1, 2008 · Information geometry emerged from studies on invariant properties of a manifold of probability distributions. It includes convex analysis and its duality as a …

WebThe concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two ... reared for meatWebIn 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 ... reared food meaningWebDec 1, 2013 · In 2011, Q. Xia [10] gave the equivalent conditions of locally dually flat (α, β)-metrics on a manifold with dimension n ⩾ 3. To find more locally dually flat Finsler metrics and tell their construction, we study the equivalent equation (1.2) and get the following equivalent equation ( F 2 ) y l x k = ( F 2 ) x l y k , whose proof is given ... reared food definitionWebFeb 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 … reared fishWebApr 1, 2014 · The issue of determining whether a metric g is a Hessian metric was raised in [8], [1] in the language of g-dually flat connections. They posed the following basic questions: Problem 1. Let (M, g) be a Riemannian manifold, does there always exist a dually flat structure on M, i.e. a pair of g-dual flat, torsion-free affine connections? … reared head backhttp://ccj.pku.edu.cn/sxjz/CN/10.11845/sxjz.2024082b reared headWebJun 4, 2024 · 2024 GMC Sierra 3500HD. The 2024 GMC Sierra 3500HD is mechanically identical to its 2024 Chevy Silverado 3500HD cousin. GMC Sierra 3500HD dual-rear … reared hindi meaning