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
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