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Information Geometry via the Q-Root Transform
Levin Maier
https://arxiv.org/abs/2603.20081 https://arxiv.org/pdf/2603.20081 https://arxiv.org/html/2603.20081
arXiv:2603.20081v1 Announce Type: new
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 \( \ell^q \)-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--\v{C}encov \(\alpha\)-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}.
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Gradualist descriptionalist set theory
David Simmons
https://arxiv.org/abs/2603.27077 https://arxiv.org/pdf/2603.27077 https://arxiv.org/html/2603.27077
arXiv:2603.27077v1 Announce Type: new
Abstract: We introduce a formal language GDST (gradualist descriptionalist set theory) with a family of interpretations indexed by ordinals, as well as a sublanguage NMID (the language of not necessarily monotonic inductive definitions), and show that the assertion that all propositions in NMID have well-defined truth values is equivalent to the existence for each $k \in \mathbb N$ of a sequence of ordinals $\eta_0 < . . . < \eta_k$ such that for each $i < k$, $\eta_i$ is $\eta_{i 1}$-reflecting, a notion we introduce which implies being $\Pi_n$-reflecting for all $n \in \mathbb N$ (and in particular being admissible and recursively Mahlo).
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