Information-Geometric Framework

• Information-Geometric Framework is a mathematical approach that endows probability distributions with differential-geometric structures like manifolds, metrics, and geodesics.
• It employs the Fisher–Rao metric and divergence functions to construct dual affine connections that underpin statistical inference, optimization, and learning theory.
• The framework extends to infinite-dimensional models and operator theory, providing unified tools for applications in deep learning, thermodynamics, and quantum theory.

An information-geometric framework systematically endows families of probability distributions and related structures with differential-geometric structures: manifolds, metrics, affine connections, curvature, and geodesics. Originally established for statistical models, modern developments have extended its scope to operator theory, thermodynamics, deep learning, non-equilibrium processes, quantum theory, and infinite-dimensional models. At its core, the framework leverages the Fisher–Rao metric, dual connections, and divergence functions—especially those that are invariant under sufficient statistics and model symmetries—to furnish canonical geometric and invariance properties directly pertinent to statistical inference, optimization, model selection, learning theory, and physical modeling.

1. Statistical Manifolds, Fisher–Rao Metric, and Divergence Construction

A parametric model $$\mathcal{M} = \{ p(x; \theta) : \theta \in \Theta \subset \mathbb{R}^d \}$$ is viewed as a $d$-dimensional smooth manifold, each point representing a probability distribution. Local charts are provided by $\theta \to p(x;\theta)$. Tangent vectors at $p_\theta$ correspond to score functions $$\partial_i \ell(\theta; x) = \partial/\partial \theta^i \log p(x;\theta)$$ (Nielsen, 2018).

The canonical Riemannian metric is the Fisher–Rao metric:

$$g_{ij}(\theta) = \mathbb{E}_{X \sim p(\cdot;\theta)}[\partial_i \ell(\theta;X) \partial_j \ell(\theta;X)] = -\mathbb{E}_{X \sim p(\cdot;\theta)}[\partial_i \partial_j \ell(\theta;X)]$$

This metric quantifies local statistical distinguishability and is the only (up to scale) metric invariant under sufficient statistics and congruent Markov morphisms (Čencov’s theorem) (Nielsen, 2018, Felice et al., 2017).

More generally, a divergence function $D(\theta:\theta')$ (vanishing only on the diagonal and sufficiently smooth) induces not only the metric but also (via Eguchi’s formalism) a family of dual affine connections: \begin{align*} g_{ij}(\theta) &= -\partial_i \partial_j' D(\theta:\theta') |{\theta' = \theta} \ \Gamma{ij,k}(\theta) &= -\partial_i \partial_j \partial_k' D(\theta:\theta') |{\theta' = \theta} \ \Gamma^*{ij,k}(\theta) &= -\partial_k \partial_i' \partial_j' D(\theta:\theta') |_{\theta' = \theta} \end{align*} Commonly, $D$ is chosen as the Kullback–Leibler divergence or a Bregman/f-divergence (Nielsen, 2018).

2. Dual Connections, Curvature, and Dually Flat Structure

Information geometry is distinguished by its dual affine connection structure $(\nabla, \nabla^*)$ with deep ties to statistical properties. The exponential (e-) connection $\nabla$ is flat in natural parameters $\theta$, while the mixture (m-) connection $\nabla^*$ is flat in mean-parameters $\eta$. The Legendre duality between $\theta$ and $\eta = \mathbb{E}_{p_\theta}[t(X)]$ is fundamental, and the Hessian of the cumulant-generating function (the log-partition) is the Fisher metric (Nielsen, 2018, Erb et al., 2020).

Amari’s $\alpha$-connections interpolate between $(\nabla, \nabla^*)$:

$$\nabla^{(\alpha)} = \frac{1 + \alpha}{2} \nabla + \frac{1 - \alpha}{2} \nabla^*$$

The dually flat structure (both connections flat) uniquely characterizes exponential families and mixture families (Nielsen, 2018, Anaya-Izquierdo et al., 2012). In such cases, global affine charts exist for both coordinates:

• e-geodesics are straight lines in $\theta$,
• m-geodesics are straight in $\eta$.

Sectional curvatures and higher-order invariants are expressed in terms of the skewness (cubic) tensor and curvature derived from the underlying divergence (Nielsen, 2018, Gauvin, 5 Mar 2025).

3. Generalizations: Nonparametric, Infinite-Dimensional, and Operator-Level Geometry

Information geometry extends to infinite-dimensional statistical models. A non-parametric manifold $$M = \{f : \mathbb{R}^n \to \mathbb{R}^+ : \int f(x) dx = 1\}$$ admits an infinite-dimensional Fisher–Rao structure with tangent spaces $T_fM$ consisting of mean-zero functions. Recent work introduces orthogonal tangent-space decompositions $T_fM = S \oplus S^\perp$ for an observable covariate subspace $S \subset T_fM$, yielding a finite-dimensional Covariate Fisher Information Matrix (cFIM) $G_f$. The trace $H_G(f) = \mathrm{Tr}(G_f)$ measures the total explainable curvature and yields Cramér–Rao bounds and intrinsic dimension estimators, directly connecting the Fisher geometry to semi-parametric efficiency and the manifold hypothesis in high-dimensional data (Cheng et al., 25 Dec 2025).

In operator-theoretic settings—e.g., imaging operators $$A : \mathcal{H}_{\mathrm{obj}} \to \mathcal{H}_{\mathrm{data}}$$—the normalized squared singular spectrum $p_i = \sigma_i(A)^2 / \sum_j \sigma_j(A)^2$ is mapped to a point on the simplex $\Delta^{N-1}$, and the Fisher–Rao metric is imposed. The resulting geometry is invariant under unitary conjugation and scaling, has constant positive sectional curvature ($+1$), and yields closed-form geodesics and distances. Operator composition induces boundary attraction and nonlinear reweighting, but isometric transport is preserved only by spectrally uniform operators (Wood, 5 Jan 2026).

4. Geometric Methods in Inference, Optimization, and Learning Theory

Information geometry underpins core results in statistical estimation:

• The Cramér–Rao bound is realized as the inverse Fisher metric, bounding the mean-square error of unbiased estimators,
• Bayesian Cramér–Rao bounds and deterministic CRLBs arise from the metric induced by an augmented divergence incorporating the prior, with posterior averaging yielding the Bayesian bound (Kumar et al., 2018),
• The Barankin bound is obtained by considering a family of quadratic forms induced by test-points, with variance lower bounds interpreted as Riemannian steepness (gradient norm) in the chosen information-geometric metric.

Optimization algorithms exploit natural gradient flow, i.e., steepest ascent in the Fisher metric, for parameter adaptation invariant under reparametrization (Akimoto et al., 2012). The Information-Geometric Optimization (IGO) framework unifies PBIL, rank-$\mu$ CMA-ES, cross-entropy, and fitness-proportional algorithms by employing stepwise natural gradient ascent of rank-encoded objective transforms, guaranteeing monotonic quantile or expectation improvement for step sizes $\delta t \leq 1$ (Akimoto et al., 2012).

Deep learning generalizations, such as the Variational Geometric Information Bottleneck (V-GIB), introduce geometric regularization into representation learning: the utility $U(\phi) = I(\phi(X); Y) - \beta \mathcal{C}(\phi)$ trades off informativeness and geometric simplicity via explicit embedding curvature penalties and intrinsic-dimension estimation. This yields nonasymptotic generalization bounds controlled by intrinsic geometric complexity rather than ambient dimension and achieves improved generalization-particularly under data scarcity-by enforcing geometric coherence of the learned manifold (Katende, 4 Nov 2025).

5. Applications across Physics, Thermodynamics, and Quantum Theory

Information geometry provides a unifying geometric infrastructure for modeling complexity, phase transitions, and chaos in physical and statistical-mechanical systems:

• In the Information Geometrodynamical Approach to Chaos (IGAC), dynamical macrostates are probability distributions on a statistical manifold with Fisher metric; geodesics, curvature, Jacobi fields, and information-geometric entropy (IGE) quantify sensitivity, chaos, and complexity (Cafaro, 2016, Ali et al., 2018, Felice et al., 2017).
• In thermodynamics, the interplay between Fisher–Rao (statistical) and Otto–Wasserstein (transport) metrics produces rigorous lower bounds on entropy production, quantifies optimal protocols as geodesics, and links entropy decay to both statistical distinguishability and mass transport cost. Modern non-equilibrium relations such as thermodynamic uncertainty relations and speed limits are interpreted as geometric inequalities (Ito, 2022).

In quantum theory, entanglement, coherence, and nonlocality are unified via multi-affine geometric frameworks: the mismatch of dual connections leads to holonomies and Berry-phase-type areas, directly realizing quantum interference and the Tsirelson bound for nonlocal correlations (Gauvin, 5 Mar 2025). Sharp distributions act as sources of curvature, and projective measurements correspond to local curvature pinching. These structures merge classical and quantum statistical geometry via divergence-induced connection algebras and multi-affine holonomy.

6. Simplex-Based, Computational, and Nonstandard Geometric Frameworks

Classical information geometry's manifold-centric approach is often inadequate for statistical models with varying dimension, mixed support, or likelihoods that concentrate on boundaries. Computational information geometry advocates the use of the probability simplex $\Delta^k$ as a global, universal geometric object that houses all submodels, their boundaries, and mixtures in a single convex structure. The Fisher metric, dual connections, and divergences are encoded in closed form throughout the simplex; models grow, shrink, or degenerate by moving onto faces of lower effective dimension (Anaya-Izquierdo et al., 2012). This geometric viewpoint enables:

• Model selection and uncertainty quantification via geodesic and curvature analysis in $\Delta^k$,
• Efficient boundary handling,
• Simultaneous embedding of arbitrary models for model averaging and information criterion computation,
• Precise control on errors via discretization.

In compositional data analysis, the information-geometric perspective generalizes Euclidean methods (clr, ilr, Aitchison distance) and supplies natural interpretations and extensions for entropy, KL-divergence, and amalgamation monotonicity, all defined on the manifold structure of the simplex with Fisher metrics and dual affine connections (Erb et al., 2020).

Semi-supervised frameworks such as GIGP leverage information-geometric concepts of divergence minimization, global context modeling, and explicit geometric priors (via moment- and invariance-matching losses) to enhance learning from limited labeled and abundant unlabeled data, aligning feature distributions and enforcing anatomical priors via geometric constraints (Yu et al., 12 Mar 2025).

7. Unification, Uniqueness, and Structural Invariance

At the highest level, information geometry offers a mathematically rigorous, coordinate-free, and invariant framework grounded in the following structural correspondences:

• Points = distributions or operators (= elements of a statistical manifold or geometry-induced simplex),
• Tangents = score directions,
• Metric = Fisher–Rao (statistical distinguishability),
• Connections = dual affine parallelisms (encoding inference and mixture geometries),
• Divergences = Bregman, KL, $\alpha$-divergences (controlling geodesics, curvature, information loss),
• Curvature = complexity, chaos, and physical criticality,
• Flows/geodesics = optimal inference, thermodynamic efficiency, or learning trajectories.

The entire framework is sustained by the requirement of invariance under sufficient statistics and congruent mappings; consequently, the Fisher–Rao metric and Amari's $\alpha$-family of dual connections emerge as uniquely distinguished structures for encoding geometry in both finite- and infinite-dimensional statistical models (Nielsen, 2018, Cheng et al., 25 Dec 2025, Katende, 4 Nov 2025, Erb et al., 2020). Through these objects, information geometry provides the foundational language to connect inference, learning, physics, and applied data science with precise, unifying geometric principles.