Potential functions in information geometry via bi-forms

Abstract: In this paper we develop a general framework for potentials on Lauritzen manifolds, namely smooth manifolds equipped with a pseudo-Riemannian metric and a pair of conjugate affine connections that may have non-vanishing torsion. We show how the theory of bi-forms accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions in a cohomological framework. Within this formalism, we construct a canonical contrast bi-form on dually curvature-free Lauritzen manifolds and establish its principal structural properties. Several illustrative examples are analysed.

• The paper introduces a unified cohomological framework for constructing potential functions on Lauritzen manifolds using differential bi-forms.
• The methodology generalizes contrast functions by incorporating pre-contrast and super-contrast functions to encode metric and full connection data.
• The framework recovers known results in finite-dimensional state spaces and Lie groups, offering new insights for torsion-full and quantum geometries.

Potential Functions in Information Geometry via Bi-forms

Introduction and Motivation

This work provides a unified, geometric framework for the construction of potential functions on Lauritzen manifolds using the theory of differential bi-forms. Lauritzen manifolds, in this context, refer to smooth manifolds endowed with a pseudo-Riemannian metric and a pair of conjugate affine connections, not necessarily torsion-free. This generalizes the classical statistical manifold setting to include structures relevant for non-commutative, quantum, and torsion-full geometries observed in modern information geometry. The central contribution is a cohomological formalism for potential functions—contrast, pre-contrast, and super-contrast functions—realized intrinsically as bi-forms and equipped with Cartan-like bicomplex structures. This enables a comprehensive description of the metric-affine geometry underlying classical, quantum, and more exotic statistical structures.

Geometric and Algebraic Foundations

The paper lays a precise foundation through the careful development of geometric structures on the Cartesian product $M \times M$ of a smooth manifold $M$. The authors construct canonical almost product structures via left and right projections, inducing two commuting integrable distributions and corresponding Frölicher-Nijenhuis bracket relations. These structures yield a bicomplex on differential forms over $M \times M$, where left ($\mathrm{d}^L$) and right ($\mathrm{d}^R$) exterior differentials act as anti-commuting cohomological operators. Vector fields, forms, and all natural geometric data on $M$ are transported to $M\times M$ via these bicomplex structures, which is critical for the subsequent analytic machinery.

Bi-forms are rigorously defined as sections of $$((\bigwedge^p \pi_L^* T^*M) \otimes (\bigwedge^q \pi_R^* T^*M)) \to M\times M$$, enabling a blockwise multilinear perspective that is locally and globally compatible with the tensor calculus on $M$. A duality operation—induced by the flip automorphism—allows systematic interchange of left- and right-structures and relates the dual statistical structures at a categorical level.

Statistical Potentials and the Unified Bi-form Formalism

The classical framework, where symmetric divergence (contrast) functions generate the Fisher metric and dual affine connections, is subsumed and generalized. The authors consider three classes of statistical potentials:

• Contrast functions: symmetric two-point functions on $M\times M$ whose Hessian yields a metric and whose third derivatives define a pair of torsion-free conjugate connections.
• Pre-contrast functions: fiberwise linear potentials on $M$0 encoding metric and a torsion-free connection; their partial derivatives need not be fully symmetric.
• Super-contrast functions: fiberwise bilinear (1,1)-bi-forms on $M$1, naturally encoding metric and full connection data, including possible torsion.

All of these are subsumed in the theory of (1,1)-bi-forms ($M$2). The metric is recovered as the diagonal restriction $M$3, and the affine connections via differential geometric operations (Lie derivatives), with precise formulas that are consistent with the classical approaches (Eguchi, Henmi-Matsuzoe, Khan-Zhang).

Strong results characterize when a Lauritzen geometry arises from a contrast bi-form, with a full hierarchy:

• If the statistical structure is a SMAT (statistical manifold with torsion), the bi-form must be left-exact: $M$4 for an appropriate $M$5-bi-form $M$6.
• If the manifold is a true statistical manifold (both connections torsion-free), the bi-form is bi-exact: $M$7 for some $M$8-bi-form (which recovers Eguchi's canonical contrast function).

The cohomology and homotopy properties of bi-forms are fully developed: using strongly convex neighborhoods (in the sense of Moretti) and geodesic interpolation, the authors construct left and right homotopy operators ($M$9) that allow unique "splittings" of bi-forms into exact/antiexact components. This gives a canonical method for decomposing potentials and extracting the essential geometric data (metric and connection), modulo irrelevant higher-order terms. These operators are shown to be explicitly "statistical" in the sense that they strictly increase the order of vanishing along the diagonal, ensuring that all meaningful geometric data remains in the (bi-)exact component.

Canonical Potentials on Dually Curvature-Free Lauritzen Manifolds

For dually curvature-free Lauritzen manifolds—those where both connections are flat but possibly with independent torsions—the paper constructs a canonical contrast bi-form using parallel frames. This bi-form enjoys several notable properties:

• It is globally left-exact in the partially flat case, with canonical pre-contrast function coinciding with Henmi-Matsuzoe’s construction.
• In the dually flat (zero torsion) case, the construction reproduces Ay-Amari’s canonical divergence.
• The solution bi-form can be expressed in terms of parallel transport along geodesics: for vector fields $M \times M$0 and points $M \times M$1,

$M \times M$2

where $M \times M$3 is the parallel transport along the geodesic from $M \times M$4 to $M \times M$5.

Moreover, when the underlying manifold has a group structure (as for Lie groups with Cartan-Schouten connections), the solution bi-form exhibits (diagonal) left- or right-invariance, matching the symmetries of the connections.

Concrete Realizations and Examples

Two classically significant families are analyzed in detail:

• Faithful states on finite-dimensional $M \times M$6-algebras: The probability simplex and quantum state spaces are realized as open submanifolds, and the formalism recovers both the classical Fisher-Rao metric and quantum monotone metrics for operator monotone functions $M \times M$7. The solution bi-form is shown to recover canonical divergences (e.g., Kullback-Leibler, von Neumann-Umegaki relative entropy, BKM divergence), and cases with non-trivial torsion are characterized by the left-exact but not bi-exact forms, in agreement with nonvanishing connection torsions beyond the BKM case.
• Semisimple Lie groups equipped with Killing metric and Cartan connections: The construction yields bi-invariant (for dually flat), left-invariant (for $M \times M$8), or diagonally left-invariant (for $M \times M$9) solution bi-forms, elucidating the interplay between group symmetries and statistical geometric structures. This explicit global construction demonstrates that algebraic obstructions (e.g., for the existence of flat left-invariant connections) do not manifest if torsion is allowed.

Structural and Cohomological Implications

A central achievement is the unification of contrast and pre-contrast functions within the bicomplex framework of bi-forms. The cohomological perspective provides precise necessary and sufficient conditions for when a metric-affine structure corresponds to (bi-)exact forms, clarifies the redundancy in higher-order jet data of bi-forms (statistical equivalence modulo vanishing of order $\mathrm{d}^L$0 at the diagonal), and introduces canonical ways of integrating and normalizing statistical potentials. This framework is highly compatible with the needs of both classical and quantum information geometry, and extends naturally to settings in which non-commutative or torsion-full connections arise in statistics and physics.

Conclusion

The systematic development of statistical potentials via bi-forms provides a formal geometric infrastructure that encompasses and extends the full range of potential functions used in information geometry. The identification of canonical cohomological and bicomplex operations, the detailed construction of homotopy/projector decompositions, and the explicit treatment of canonical models in both commutative and non-commutative examples demonstrate the effectiveness and generality of the formalism.

Several open directions remain, including the precise characterization of which Lauritzen structures always admit a contrast bi-form in the most general torsion-full, curved setting, and a more detailed exploration of the statistical interpretation of tangent vectors in torsion-full geometries. The developed framework suggests broader applications for potential-theoretic methods in differential geometry, quantum information, and possibly the study of geometric flows and deformation quantization.