Covariate Restrictions of the Fisher–Rao Metric: Spectral Geometry, Semiparametric Efficiency, Tracial Categorification, and the Path-Space Geometry of Diffusions

The Fisher–Rao metric on the infinite-dimensional manifold of densities is a metric functional
whose inversion and spectral analysis are widely regarded as intractable. A productive response
is to restrict the metric to a finite-dimensional, statistically meaningful subspace of the tangent
space and to study the resulting Gram matrix. We give a clean operator-theoretic formulation of
this idea through a covariate score operator SA attached to an arbitrary admissible observable
frame A; its Gram matrix Gf (A) = S∗
ASA is the covariate Fisher information matrix (cFIM),
which simultaneously generalizes the coordinate (translation) frame, exponential-family feature
frames, and vector-field/group frames, and recovers a recently proposed orthogonal-decomposition
framework as a special case. We then carry this calculus through geometry, category theory,
statistics, and stochastic dynamics. Foundations: the scalar invariant (“G-entropy”) is the
classical Fisher information functional J(f) =
R
∥∇ log f∥2f; the assignment f 7→ Gf (A) is
functorial under sufficient statistics, monotone under coarse-graining, and admits a coend
description. Categorification: we resolve the integer/real type mismatch that blocks a naive
Euler-characteristic lift by categorifying in a tracial von Neumann (W∗) category (Mf , τf ) =
(L∞(X), Ef ), in which the metric is the GNS pairing, the G-entropy is the real-valued trace of a
positive frame observable, the Pythagorean law is trace additivity, functoriality is a Pimsner–Popa
contraction, and the dual connections together with the Amari–Chentsov cubic tensor are the
cumulants of the free-energy potential, the cubic tensor being the leading A∞ obstruction to dual
symmetry. Manifold Hypothesis: for a density concentrated within scale σ of a d-manifold the
cFIM has n − d eigenvalues of order σ−2 (normal bundle) and d bounded eigenvalues (tangent
bundle), so intrinsic dimension is the number of small eigenvalues—reversing a published
prescription and matching the diffusion-model literature—and we prove exact finite-sample
recovery b d = d with probability 1 − 2e−cN once N ≳ n − d, with an eigenvalue/subspace central
limit theorem and a consistent spectral-gap test. Efficiency and design: a frame-selection principle
replaces an unjustified “geometric alignment” postulate—the restricted geometry attains the
semiparametric bound exactly when the frame spans the efficient score—and an Eckart–Young
theorem in the Fisher metric identifies the optimal frame under a dimension budget, with the
restricted inverse cFIM as natural-gradient preconditioner. Dynamics: a generalized de Bruijn
identity shows the anisotropic G-entropy is the dissipation rate of relative entropy along any
Fokker–Planck flow, with Bakry–´Emery exponential decay; and via Girsanov’s theorem the laws
of Itˆo diffusions form an infinite-dimensional Fisher–Rao manifold whose metric is the expected
time-integral of the squared drift perturbation, onto which the entire apparatus lifts, the marginal
projection recovering the de Bruijn dissipation. We close with quantum, optimal-transport,
and reproducing-kernel extensions and four open problems linking modular theory, multiscale
1
SDEs, semiparametric drift estimation, and nonequilibrium entropy production to the framework.
A suite of controlled numerical experiments, packaged as a reproducible notebook, confirms
the central predictions—the eigenvalue split and dimension-from-small-eigenvalues rule, exact
recovery at N ∼ n − d with the eigenvalue central limit theorem, the partially-linear-model
efficiency gap as a law of total variance, the de Bruijn dissipation identity to machine precision
(and its persistence under irreversible circulation, with fivefold acceleration), the Fisher-to-Otto
ratio k on Hermite modes, the SLD/Kubo–Mori quantum split, and the Cartan/Amari–Chentsov
identification.