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相关概念视频

Gradient and Del Operator01:14

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The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
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镜子下降和指数梯度算法使用痕迹形式度.

Andrzej Cichocki1,2,3,4, Toshihisa Tanaka3, Frank Nielsen5

  • 1Systems Research Institute of Polish Academy of Science, Newelska 6, 01-447 Warsaw, Poland.

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概括
此摘要是机器生成的。

这项研究提出了新的镜像下降 (MD) 和使用通用的通用指数梯度 (GEG) 算法. 这些方法通过适应复杂的几何形状,提供了更好的收性和稳定性.

关键词:
(q,κ) - - 代数中的一个.布雷格曼分歧是什么意思里姆尼安优化的优化.变形的对数是对数的变形.一般化的指数梯度渐变.信息几何学信息几何学镜子下降的下降自然梯度的自然梯度.

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科学领域:

  • 优化理论 优化理论
  • 信息几何学信息几何学
  • 机器学习 机器学习

背景情况:

  • 镜像下降 (MD) 和通用指数梯度 (GEG) 是基本的优化算法.
  • 经典方法经常与消失/爆炸梯度和非欧几里德几何学作斗争.
  • 一般化的率为定义分歧和指标提供了一个灵活的框架.

研究的目的:

  • 引入MD和GEG算法的统一框架,基于概括的痕迹形式.
  • 为了证明这些新算法的改进的收性和稳定性特性.
  • 揭示这些方法与自然梯度下降相连接的信息几何基础.

主要方法:

  • 通过变形的对数来导出MD和GEG算法从痕迹形式的值.
  • 对收行为和梯度强度的分析.
  • 调查与阿玛里的自然梯度和信息几何结构的联系.
  • 适用于特定的家族 (Tsallis,Kaniadakis等) 为了定义里曼的指标.

主要成果:

  • 开发一类广泛的MD和GEG算法,以增强的融合和稳定性.
  • 为各种梯度更新规则 (附加,乘法,自然) 建立统一的几何基础.
  • 证明不同的值诱导不同的里曼度量,保持统计几何学.
  • 可调节的参数允许自适应的几何选择,以提高优化.

结论:

  • 拟议的框架将第一阶段优化方法统一在一般化的布雷格曼分歧下.
  • 的选择决定了底层的里曼度量和双重几何结构.
  • 与经典欧几里德优化相比,这些通用方法提供了增强的适应性和稳定性.