Reframing of Information Geometry via Symmetric Teleparallel Gravity

Abstract: Information geometry has traditionally been formulated within the framework of Riemannian geometry and dual affine connections. In this work, we reframe this foundational structure by introducing the geometric machinery of symmetric teleparallel gravity. By requiring both curvature and torsion to vanish globally on the statistical manifold, we demonstrate that the fundamental properties of the information space can be entirely encoded into the non-metricity tensor. This approach allows us to distinguish the general $ξ$-parameterized space from the $θ$- (or $η$-) parameterized space, mirroring the relationship between conventional general relativity and symmetric teleparallel gravity. Specifically, the $θ$- or $η$-coordinates emerge as the special coordinates in the coincident gauge, where the connection coefficients vanish.

• The paper demonstrates that enforcing zero curvature and torsion recasts the geometry of statistical manifolds in terms of a symmetric non-metricity tensor.
• It distinguishes between general and natural parameterizations, analogous to the contrast between general relativity and symmetric teleparallel gravity.
• The work leverages Stueckelberg fields to restore gauge invariance, offering new tools for coordinate-invariant analysis in complex statistical models.

Reframing Information Geometry via Symmetric Teleparallel Gravity

Overview

This paper establishes a formal correspondence between the frameworks of information geometry (IG) and symmetric teleparallel gravity (STG). By rigorously enforcing the vanishing of both curvature and torsion on the statistical manifold, the work demonstrates that the geometric structures underlying IG can be comprehensively recast in terms of the non-metricity tensor, a central concept in STG. This approach leads to a principled distinction between general parameterizations and the so-called "natural" parameterizations underlying dually flat spaces in IG, analogous to the contrast between general relativity (GR) and STG in gravitational theories.

Symmetric Teleparallel Gravity and Non-Metricity

STG provides an alternative to the standard metric-affine description of gravity, positing a spacetime with globally vanishing curvature and torsion but non-trivial non-metricity. The non-metricity tensor $Q_{kij} = \nabla_k g_{ij}$, as the sole non-vanishing geometric object, encodes changes in the metric under parallel transport in a flat underlying manifold. The formalism permits the global adoption of a "coincident gauge"—a coordinate system in which all affine connection components vanish identically on the manifold. Under coordinate transformations parameterized by St\"uckelberg fields, this gauge choice enables an unambiguous separation of metric and connection degrees of freedom.

The decomposition of a general torsion-free affine connection into Levi-Civita and disformation components underscores the distinctiveness of STG. The connection can be expressed as

$${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$

with the disformation $L^k{}_{ij}$ determined entirely by non-metricity, highlighting the primacy of non-metricity in this geometrical setup.

Information Geometry: Dually Flat Spaces and $\alpha$-Connections

IG treats the space of probability distributions, typically the exponential family, as a statistical manifold equipped with the Fisher metric and dual flat affine connections. The "dually flat" structure $(\mathcal{S}, g, \nabla, \nabla^\star)$ features two flat, mutually dual, torsion-free connections, which respectively admit flat coordinate systems—the $\theta$- and $\eta$-coordinates. The geometry is governed by a pair of convex functions linked via Legendre transformation, with the Fisher metric given by the Hessian of these potentials.

In conventional IG, the dualistic connections and the Amari-Centsov tensor play foundational roles, but non-metricity is rarely foregrounded. Recently, it was shown that the non-metricity tensor encodes the $\alpha$-connection structure and is directly proportional to the Amari-Centsov tensor, $Q_{ijk} = \alpha C_{ijk}$. This insight suggests a latent presence of non-metricity as a defining property of the information manifold, especially when viewed in the context of dually flat spaces.

Geometric Reframing via STG

This work reframes the foundational structure of IG using STG machinery. By requiring both curvature and torsion to vanish globally—enforcing a fully flat connection—and focusing on the coincident gauge, it establishes that all non-trivial geometry emerges from non-metricity. The paper formalizes the role of parameterizations, distinguishing general coordinates ($\xi$-coordinates) from $${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$0- (or $${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$1-) coordinates particular to dually flat structures. The $${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$2- and $${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$3-coordinates correspond precisely to the coincident gauge in STG, where connection coefficients vanish, and the metric's Hessian structure manifests via the totally symmetric non-metricity tensor.

Through St\"uckelberg's trick, the covariance of the formalism is restored, treating the coordinates associated with the coincident gauge as scalar fields under general diffeomorphisms. This allows a rigorous distinction between general parameterizations and special parameterizations yielding dually flat geometry. The connection coefficients in general coordinates $${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$4 are then derived explicitly in terms of the transformations to the coincident gauge, matching the structure known from STG.

The crucial implication is that the dually flat structure, the existence of an Amari-Centsov tensor, and the metric's Hessian property are all consequences of non-metricity taking a totally symmetric form in the coincident gauge. The framework reveals that dual flatness and the geometric underpinnings of IG are not unique to the conventional affine connection deformation viewpoint, but rather naturally arise from the constraint to vanishing curvature and torsion with non-trivial non-metricity.

Implications and Future Directions

The non-trivial identification of the statistical manifold's geometry with that of an STG spacetime offers new perspectives on the foundation of information geometry. Practically, this correspondence could yield new tools for statistical manifold analysis, potentially enabling coordinate-invariant treatment of connections and metrics in complex statistical models.

Theoretically, this reframing suggests that information-theoretic notions such as "information dissipation" may be interpreted geometrically via non-metricity, paralleling gravitational force representations in STG. The explicit use of non-metricity provides a robust geometric distinction between general and natural statistical parameterizations, hinting that further physical analogies between information flow and gravitational field representation could be explored.

Future research may leverage this geometric correspondence for new invariants or information-theoretic quantities, especially in nonequilibrium statistical physics, quantum information, or learning theory, where parameterization-invariant formulations are increasingly critical. Further, analogies with gauge theoretic restoration of symmetry (via St\"uckelberg fields) suggest possible extensions to gauge-theoretic formulations of information geometry or connections to modern approaches in quantum gravity.

Conclusion

By reframing information geometry within the structure of symmetric teleparallel gravity, this work provides a principled geometrization of the statistical manifold rooted in non-metricity. The approach clarifies the geometric origin of dual affine connections, the special status of $${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$5- and $${}^k{}_{ij} = {}^\text{LC}{}^k{}_{ij} + L^k{}_{ij},$$6-coordinates, and the fundamental role of the Amari-Centsov tensor, while establishing a concrete analogy with the trinity of gravity theories. This perspective lays the groundwork for both theoretical advances and practical tools in the rigorous analysis of statistical geometries.