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

Abstract

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.