2025/05/31 by Maier, Levin
Mathematics · Computer Science · #Mathematical Analysis and Transform Methods #Coding theory and cryptography #Advanced Data Compression Techniques
paper · pdf · doi:10.48550/arxiv.2506.00485
In this paper, we introduce ℓp-information geometry, an infinite dimensional framework that shares key features with the geometry of the space of probability densities \( Dens(M) \) on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the ℓ2-probability simplex with a noncanonical differentiable structure induced via the q-root transform from an open subset of the ℓp-sphere. This structure renders the q-root map an isometry, enabling the definition of Amari--Čencov α-connections in this setting. We further construct gradient flows with respect to the ℓ2 Fisher--Rao metric, which solve an infinite-dimensional linear optimization problem. These flows are intimately linked to an integrable Hamiltonian system via a momentum map arising from a Hamiltonian group action on the infinite-dimensional complex projective space.