Information Geometry on the ${\mathrm{\ell }}^2$-Simplex via the $q$-Root Transform

In this paper, we introduce \emph{${\mathrm{\ell }}^p$-information geometry}, an infinite dimensional framework that shares key features with the geometry of the space of probability densities $\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}(M)$ on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the \emph{${\mathrm{\ell }}^2$-probability simplex} with a noncanonical differentiable structure induced via the \emph{$q$-root transform} from an open subset of the ${\mathrm{\ell }}^p$-sphere. This structure renders the $q$-root map an \emph{isometry}, enabling the definition of \emph{Amari–Čencov $\alpha$-connections} in this setting. We further construct \emph{gradient flows} with respect to the ${\mathrm{\ell }}^2$ 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.