标题： 通过$q$-根变换在$\ell^2$-单形上的信息几何 显示英文标题

标题： Information Geometry on the $\ell^2$-Simplex via the $q$-Root Transform

摘要： 在本文中，我们引入了\emph{$\ell^p$-信息几何}，这是一个无限维框架，它与闭流形上概率密度空间\( \mathrm{Dens}(M) \)的几何结构具有关键特征，同时结合了测度值信息几何的某些方面。 我们通过从\emph{$q$-根变换}的开子集诱导的$\ell^p$-球面定义了\emph{$\ell^2$-概率单纯形}。 这种结构使得$q$-根映射成为\emph{等距变换}，从而在此设置中定义\emph{阿马里-岑科夫$\alpha$-连接}。 我们进一步相对于$\ell^2$Fisher--Rao 度量构造\emph{梯度流}，这解决了无限维线性优化问题。 这些流通过来自无限维复射影空间上的哈密顿群作用产生的\emph{动量映射}与\emph{可积哈密顿系统}密切相关。 显示英文摘要

摘要： In this paper, we introduce \emph{$\ell^p$-information geometry}, an infinite dimensional framework that shares key features with the geometry of the space of probability densities \( \mathrm{Dens}(M) \) on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the \emph{$\ell^2$-probability simplex} with a noncanonical differentiable structure induced via the \emph{$q$-root transform} from an open subset of the $\ell^p$-sphere. This structure renders the $q$-root map an \emph{isometry}, enabling the definition of \emph{Amari--\v{C}encov $\alpha$-connections} in this setting. We further construct \emph{gradient flows} with respect to the $\ell^2$ Fisher--Rao metric, which solve an infinite-dimensional linear optimization problem. These flows are intimately linked to an \emph{integrable Hamiltonian system} via a \emph{momentum map} arising from a Hamiltonian group action on the infinite-dimensional complex projective space.