论文标题
从广义算术手段到测量学到汉密尔顿动态到布雷格曼的分歧
From generalized arithmetic means to geodesics to Hamilton dynamics to Bregman divergences
论文作者
论文摘要
在这里，我们研究了广义算术手段，大地测量学，拉格朗日 - 哈米尔顿动力学和布雷格曼分歧之间的一些联系。在上一篇论文中，我们对广义算术手段进行了预测解释。这项工作更概率地定向。在这里，我们进行了几何转弯，并看到一般的算术手段实际上最大程度地减少了$ \ mathbb {r}^n。$ $ $ \ m athbb {r}^n。$ y Mathbb {r}^n。$可能会在某些情况下可能会造成这些扣子的次数，这可能会造成这些指标，从而在某些情况下可以看到，这些指标可能会造成这些指标的功能。当凸函数的Hessian具有平方根的Hessian是$ \ Mathbb {r}^n。$中的差异性的雅各布，在这种情况下，我们获得了由Convex函数定义的Bregman Divergence的比较。
Here we examine some connections between the notions of generalized arithmetic means, geodesics, Lagrange-Hamilton dynamics and Bregman divergences. In a previous paper we developed a predictive interpretation of generalized arithmetic means. That work was more probabilistically oriented. Here we take a geometric turn, and see that generalized arithmetic means actually minimize a geodesic distance on $\mathbb{R}^n.$ Such metrics might result from pull-backs of the Euclidean metric in $\mathbb{R}^n.$ We shall furthermore see that in some cases these pull-backs might coincide with the Hessian of a convex function. This occurs when the Hessian of a convex function has a square root that is the Jacobian of a diffeomorphism in $\mathbb{R}^n.$ In this case we obtain a comparison between the Bregman divergence defined by the convex function and the geodesic distance in the metric defined by its Hessian.
