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 \( \ell^q \)-sphere. This choice makes the \(q\)-root transform an \emph{isometry} and allows us to construct the \(\ell^2\)- and \(\ell^q\)-Fisher--Rao geometries, including \emph{Amari--Čencov \(α\)-connections} and a \emph{Chern connection} in the \(\ell^q\)-setting. We then apply this framework to an infinite-dimensional linear optimization problem. We show that the corresponding gradient flow with respect to the \(\ell^2\)--Fisher--Rao metric can be solved explicitly, converges to a maximizer under a natural monotonicity assumption, and admits an interpretation as the geodesic flow of an \emph{exponential connection}. In particular, we prove that this \(e\)-connection is \emph{geodesically complete}. We further relate these flows to a \emph{completely integrable Hamiltonian system} through a \emph{momentum map} associated with a Hamiltonian torus action on infinite-dimensional complex projective space. Finally, inspired by the \(\ell^2\)-theory, we outline an analogous Fisher--Rao geometry for \( \mathrm{Dens}(M) \) on possibly noncompact Riemannian manifolds, showing that, with a suitable spherical differentiable structure, the square-root transform remains an \emph{isometry}.

翻译：本文提出“ℓᵖ-信息几何”这一无穷维框架，该框架兼具闭流形上概率密度空间Dens(M)几何与测度值信息几何的特征。我们通过从ℓᵠ-球的开子集出发的q-根变换，赋予ℓ²-概率单纯形非标准微分结构。这一选择使q-根变换成为等距映射，进而构建ℓ²-与ℓᵠ- Fisher-Rao几何，包括ℓᵠ-设定下的Amari-Čencov α-联络与Chern联络。随后将该框架应用于无穷维线性优化问题：证明关于ℓ²-Fisher-Rao度量的梯度流可显式求解，在自然单调性假设下收敛至极大点，并可解释为指数联络的测地流。特别地，我们证明该e-联络是测地完备的。进一步通过无穷维复射影空间上哈密顿环面作用相关的动量映射，将这些流与完全可积哈密顿系统相联系。最后受ℓ²-理论启发，我们概述了可能非紧黎曼流形上Dens(M)的Fisher-Rao几何类比，证明在合适的球形微分结构下，平方根变换仍保持等距性质。