论文标题
单数模型的双重平坦结构
The dually flat structure for singular models
论文作者
论文摘要
信息几何和相关字段中突出显示了由Amari-Nagaoka引入的双重平坦结构。但是，在实际应用中，基本的伪里曼尼亚式度量通常可能是退化的，并且在整个空间上很少定义出如此出色的几何结构。在本文中，我们提出了从拉格朗日和Legendre奇异性理论的角度来提出对某些类别模型的双重平面结构的新颖概括 - 我们引入了一种准赫西亚语流形，这些流形具有可能是退化的度量指标，并可能超出了对对称的立方张量，这超出了统计的概念和统计的概念。特别是，我们在此一般设置中建立了Amari-Nagaoka的扩展毕达哥拉斯定理和投影定理，因此，即使对于此类奇异案例，这些定理的大多数应用程序也适当合理。这项工作是由各种兴趣的动机，这些兴趣从数学物理学的弗罗贝尼斯结构到数据科学的深度学习，具有不同的背景。
The dually flat structure introduced by Amari-Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, and such an excellent geometric structure is rarely defined on the entire space. To fix this trouble, in the present paper, we propose a novel generalization of the dually flat structure for a certain class of singular models from the viewpoint of Lagrange and Legendre singularity theory - we introduce a quasi-Hessian manifold endowed with a possibly degenerate metric and a particular symmetric cubic tensor, which exceeds the concept of statistical manifolds and is adapted to the theory of (weak) contrast functions. In particular, we establish Amari-Nagaoka's extended Pythagorean theorem and projection theorem in this general setup, and consequently, most of applications of these theorems are suitably justified even for such singular cases. This work is motivated by various interests with different backgrounds from Frobenius structure in mathematical physics to Deep Learning in data science.
