发表机构
Nagoya Mathematical and Information Science Research（名古屋数学与信息科学研究）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文介绍“幂几何”研究纲领，涵盖概率分布起源与矩阵动力系统，扩展布罗克特双括号流至矩形矩阵等，通过连续幂参数插值特征值问题，为经典与量子信息处理算法提供设计原则。
AI 中文摘要
本文是对一篇原始日文著作的修正英文翻译，该著作介绍了“幂几何”，这是一个以优化理论意义上的函数的幂变换、变形和对偶性为核心的研究纲领。其两个主要动机是概率分布的起源（通过极大似然原理、有理函数参数以及与信息几何、皮尔逊系统和施瓦茨方程相关的常微分方程）以及计算特征值、奇异值和主子空间或次子空间的矩阵动力系统的构造。作者在布罗克特的双括号流的基础上，将这些系统扩展到矩形矩阵、旗流形和凸差势，并展示了连续幂参数如何在最大和最小特征值问题之间进行插值。本文考察了对偶势、主子空间流和次子空间流之间的拓扑区别，以及通过矩阵施瓦茨-里卡蒂方程的联系，旨在为经典和量子信息处理算法的设计原则提供参考。
英文摘要
This paper is a corrected English translation of an original Japanese work introducing "power geometry," a research programme centred on power transformations, deformations and dualities of functions in the sense of optimisation theory. Its two main motivations are the origin of probability distributions (via maximum-likelihood principles, rational-function parameters and ODEs linked to information geometry, the Pearson system and Schwarzian equations) and the construction of matrix dynamical systems that compute eigenvalues, singular values and principal or minor subspaces. Building on Brockett's double-bracket flows, the author extends these systems to rectangular matrices, flag manifolds and difference-of-convex potentials, and shows how a continuous power parameter can interpolate between largest- and smallest-eigenvalue problems. Dual potentials, topological distinctions between principal and minor subspace flows, and links via matrix Schwarzian--Riccati equations are examined, with a view toward design principles for classical and quantum information-processing algorithms.
Comments13 pages, English translation of the original Japanese source
