Dually Flat Manifolds

• Dually flat manifolds are smooth manifolds endowed with a Riemannian metric and two torsion-free, mutually dual flat affine connections, establishing global affine coordinate systems through a convex potential.
• Their structure facilitates the derivation of canonical divergences like the Bregman divergence and supports key geometric results such as the generalized Pythagorean theorem.
• These manifolds find applications in statistical inference, machine learning algorithms like mirror descent, and advanced fields such as toric and Kähler geometry.

A dually flat manifold is a smooth manifold equipped with a Riemannian metric and two torsion-free affine connections—the primal and the dual—that are both flat (zero curvature) and mutually dual with respect to the metric. These structures emerged from information geometry, where they underpin the differential-geometric study of exponential families, mixture families, and statistical divergences, notably the Kullback–Leibler (KL) divergence. Dually flat geometry enables the development of powerful algorithms based on Bregman divergences, origins in Amari’s theory of statistical manifolds, and a diverse range of applications in statistics, optimization, toric geometry, and the geometry of Finsler and Hessian manifolds.

1. Fundamental Structure of Dually Flat Manifolds

A dually flat manifold $(M, g, \nabla, \nabla^*)$ consists of a Riemannian metric $g$ and a pair of torsion-free flat affine connections $\nabla$ (primal) and $\nabla^*$ (dual), satisfying the duality relation

$$X \cdot g(Y,Z) = g(\nabla_X Y, Z) + g(Y, \nabla^*_X Z)$$

for vector fields $X,Y,Z$. Flatness of both connections ensures the existence of global affine coordinate systems $\theta$ (for $\nabla$) and $\eta$ (for $\nabla^*$), related via a strictly convex potential $F(\theta)$:

• $g_{ij}(\theta) = \partial_i \partial_j F(\theta)$
• Dual coordinates: $\eta_i = \partial_i F(\theta)$
• Legendre dual potential: $$F^*(\eta) = \sup_\theta \{ \langle \theta, \eta \rangle - F(\theta) \}$$
• $\theta$ and $\eta$ provide global affine charts for $\nabla$ and $\nabla^*$ respectively.

In statistical manifolds equipped with a divergence function $D(P:Q)$, the Riemannian metric is often derived as the Hessian of $D$ at the diagonal: $$g_{ij}(\theta) = \left. \frac{\partial^2}{\partial \theta_1^i \partial \theta_2^j} D(p(\cdot; \theta_1)\!:\!p(\cdot; \theta_2)) \right|_{\theta_1 = \theta_2 = \theta}$$ KL divergence induces a dually flat structure on exponential and mixture families, with Bregman divergence as the canonical divergence (Nielsen et al., 2018).

2. Canonical Divergences and Dual Geometry

The canonical divergence (Bregman-type divergence) on a dually flat manifold is given by

$$D(p:q) = F(\theta(p)) - F(\theta(q)) - (\theta(p)-\theta(q))^T \nabla F(\theta(q))$$

with a dual divergence in $\eta$-coordinates using the Legendre dual $F^*(\eta)$. Key properties include:

• Asymmetry: $D(p:q) \neq D(q:p)$ in general.
• Nonnegativity: $D(p:q) \ge 0$ with equality iff $p=q$.
• Geometric interpretation: The divergence generalizes squared Euclidean distance, with the "triangular relation" functioning as a non-Euclidean law of cosines (Nishiyama, 2018).

The "generalized Pythagorean theorem" holds: for an $\nabla$-geodesic and an $\nabla^*$-geodesic meeting orthogonally,

$D(p:q) + D(q:r) = D(p:r)$

as seen in the three-point Bregman identity (Nielsen, 2019).

3. Model Classes and Explicit Constructions

Exponential and Mixture Families

• Exponential families: $$p(x;\theta) = \exp(\langle t(x), \theta \rangle + k(x) - F(\theta))$$, with cumulant generating function $F(\theta)$ strictly convex.
• Mixture families: $$m(x;\eta) = \sum_{i=1}^D \eta_i p_i(x) + (1 - \sum_{i=1}^D \eta_i) p_0(x)$$, with negative entropy $G(\eta) = \int m(x;\eta) \log m(x;\eta) dx$ as the generator.

KL divergence between two members induces Bregman divergences in parameter space. For mixture or exponential families lacking closed-form generators (e.g., non-discrete or continuous supports), Monte Carlo estimators yield stochastic strictly convex approximations that under natural assumptions converge almost surely to the true generator (Nielsen et al., 2018).

Finsler and (α,β)-Metrics

The concept extends to Finsler geometry, where an $(\alpha,\beta)$-metric is locally dually flat if and only if it admits a representation through a Riemannian metric and 1-form satisfying specific duality conditions and a characterization PDE for the metric function. The β-deformation method systematically generates all dually flat $(\alpha,\beta)$-metrics from dually flat Riemannian metrics plus dually related 1-forms (Yu, 2013). This creates large families of nontrivial dually flat Finsler metrics, encompassing and generalizing the Randers case.

Dually Flat Metric Construction Table

Model Type	Potential Generator	Flat Coordinates
Exponential family	Cumulant function $F$	Natural $\theta$, Expectation $\eta$
Mixture family	Negative entropy $G$	Mixing weights $\eta$, Cross-entropy $\theta$
(α,β)-metric (Finsler)	Metric function φ(b²,s)	Riemannian + 1-form variables

4. Geometric and Algorithmic Implications

Geodesics and Orthogonality

• $\nabla$-geodesics: straight lines in $\theta$-space.
• $\nabla^*$-geodesics: straight lines in $\eta$-space.
• Orthogonality: For tangent vectors $u, v \in T_qM$, $g_q(u, v^*) = 0$ translates to primal–dual affine coordinates being orthogonal (Nielsen, 2019).
• The geometry accommodates both excess and defect in geodesic triangle angle sums, with only Euclidean (self-dual) cases preserving the classical angle sum $\pi$.

Optimization and Statistical Algorithms

The straight-line structure in dual coordinates yields efficient optimization algorithms:

• Geodesic descent: m-geodesic updates in exponential families recover the exact maximum likelihood estimator in a single step. e-geodesic updates correspond to practical mirror descent and exponentiated-gradient methods (Omiya et al., 10 Dec 2025).
• Bregman-based learning: All Bregman geometry algorithms (k-means, Voronoi diagrams, minimum enclosing balls, etc.) transfer directly, with theoretical guarantees of nonnegative divergences and consistency ensured by the convex Bregman generators (Nielsen et al., 2018).

Divergences and Inequalities

Multiple divergences arise—canonical (Bregman), affine symmetric (Jeffreys), Jensen-Shannon, Bhattacharyya, and α-divergences—each with explicit representation in dually flat settings. Inequalities between divergences, such as Lin's inequality ($\frac14 D_J \geq D_{JS}$), generalize cleanly (Nishiyama, 2018), reflecting the underlying geometry.

5. Extensions: Singular Structures, Toric Geometry, and Kähler Torifications

Quasi-Hessian and Singular Models

In degenerate settings where the metric is not globally nonsingular, quasi-Hessian manifolds generalize the classical dually flat theory by replacing the dual connections by flat coherent tangent bundles, along with a well-defined symmetric cubic tensor and an extended notion of canonical divergence. Extended Pythagorean and projection theorems persist in singular cases relevant for singular learning theory, deep networks, and certain Frobenius-type structures (Nakajima et al., 2020).

Toric and Kähler Correspondences

Dually flat manifolds naturally lift to Kähler toric manifolds (“torification”), generalizing the Delzant construction. Specifically, every regular Kähler toric manifold corresponds uniquely to a dually flat manifold with global dual coordinates, and vice versa. The toric Kähler geometry recovers the convex polytope structure (moment images), and the Bregman divergence structure extends up to the polytope boundary (Fujita, 2023, Molitor, 2021, Figueirêdo et al., 2023). In dimension one, all toric dually flat manifolds instantiate standard complex space forms: spheres, projective spaces, and hyperbolic planes.

Twisted/Warred Product Geometries

Product-type constructions are classified: any twisted product $M = B \times_b F$ is dually flat if and only if $B$ is dually flat and $F$ has constant sectional curvature, under suitable flatness or conformality criteria on the twisting function (Diallo et al., 2014).

6. Applications Across Domains

Dually flat manifolds underpin major areas:

• Information geometry: Geometry of exponential and mixture families, statistical inference, clustering, and divergence minimization (Nielsen et al., 2018, Omiya et al., 10 Dec 2025).
• Optimization theory: Theoretical grounding for mirror descent, exponentiated gradient, and geodesic-based algorithms (Omiya et al., 10 Dec 2025).
• Machine learning: Geometry of estimation for singular models and deep learning architectures with degenerate Fisher metrics (Nakajima et al., 2020).
• Signal processing and statistics: Escort probabilities and α-geometry for non-additive statistics and α-divergence-based computations (Ohara et al., 2010).
• Mathematical physics: Frobenius manifolds, Dubrovin duality, and WDVV equations in connection with bi-flat and dually flat structures (Proserpio et al., 9 Dec 2025).
• Geometry and topology: Construction of Finsler and Hessian manifolds, Delzant polytopes, and Kähler toric manifolds (Fujita, 2023, Molitor, 2021, Figueirêdo et al., 2023).

7. Advanced Structures and Current Research Directions

Modern developments include:

• Monte Carlo information geometries: Enabling practical computation on complex or intractable distribution families via MC estimators that define strict, convex, smooth generators almost surely, preserving dually flat geometry (Nielsen et al., 2018).
• Contact geometry perspectives: Interpreting dually flat structures as Legendre manifolds within contact manifolds, with applications in circuit theory and nonequilibrium statistical mechanics (Goto, 2015).
• Explicit classification and singularity theory: Complete determination of dually flat $(\alpha,\beta)$-metrics and singular Hessian manifolds, including explicit degeneration loci and bifurcation behavior (Yu, 2013, Nakajima et al., 2020).
• Quantum information geometry: Embedding probability simplex as toric varieties, relating Born’s rule, Fubini–Study metric, and entanglement structure with the geometry of dually flat spaces (Molitor, 2021, Figueirêdo et al., 2023).

Ongoing research explores further relationships with tropical geometry, algebraic statistics, category-theoretic frameworks, and moduli spaces, extending the reach and mathematical richness of dually flat geometry.

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References

• Monte Carlo Information Geometry: The dually flat case (Nielsen et al., 2018)
• On dually flat $(α,β)$-metrics (Yu, 2013)
• On geodesic triangles with right angles in a dually flat space (Nielsen, 2019)
• Divergence functions in dually flat spaces and their properties (Nishiyama, 2018)
• The dually flat structure for singular models (Nakajima et al., 2020)
• Kahler toric manifolds from dually flat spaces (Molitor, 2021)
• The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry (Fujita, 2023)
• Toric dually flat manifolds and complex space forms (Figueirêdo et al., 2023)
• Minimization of Functions on Dually Flat Spaces Using Geodesic Descent Based on Dual Connections (Omiya et al., 10 Dec 2025)
• Contact geometric descriptions of vector fields on dually flat spaces (Goto, 2015)
• On dually flat general $(α,β)$-metrics (Yu, 2013)
• Dualistic Structures on Twisted Product Manifolds (Diallo et al., 2014)
• Dually flat structure with escort probability and its application to alpha-Voronoi diagrams (Ohara et al., 2010)
• Dubrovin duality for open Hurwitz flat F-manifolds (Proserpio et al., 9 Dec 2025)