Geometry-Driven Optimal Transport Fusion

• Geometry-driven optimal transport fusion is a framework where differential and topological properties of cost functions determine the structure and regularity of transport solutions.
• The approach leverages pseudo-Riemannian metrics, symplectic forms, and curvature conditions to ensure uniqueness and smoothness of Monge maps and Kantorovich measures.
• This fusion method unifies diverse OT models and informs practical applications in machine learning, fluid mechanics, and nonlinear PDE analysis.

Geometry-driven optimal transport fusion refers to a broad class of theoretical frameworks and practical methodologies in which the geometric structure underlying an optimal transport (OT) problem is not merely an ancillary consideration but becomes the organizing principle that dictates the existence, regularity, and structure of the transport, the design of fusion mechanisms, and their consequences for applications. This perspective emphasizes how differential and topological properties of the cost function, pseudo-Riemannian metrics, and associated symplectic structures fundamentally govern the behavior of Monge maps, Kantorovich couplings, and the fused objects that arise from transport-theoretic formulations. The integration of geometric tools enables both a unification of seemingly diverse OT models and deeper insights into connections with nonlinear partial differential equations, curvature conditions, and real-world applications in areas such as fluid mechanics, machine learning, and physics.

1. Differential Geometry of Optimal Transport

A central insight is that optimal transport is intrinsically a geometric problem: the transportation cost function $c(x, y)$ not only acts as a penalty function but also shapes the topology and geometry of the problem through its differential properties. A key object is the cross-difference function,

$$\delta(x, y; x_0, y_0) = c(x, y_0) + c(x_0, y) - c(x, y) - c(x_0, y_0).$$

The Taylor expansion of $\delta$ near the diagonal, $\delta^0(x_0+\Delta x, y_0+\Delta y)$, is

$$\delta^0(x_0+\Delta x, y_0+\Delta y) = h((\Delta x, \Delta y), (\Delta x, \Delta y)) + o(|\Delta x|^2+|\Delta y|^2),$$

where $h$ (the Hessian of $\delta$) is a pseudo-Riemannian metric of split signature on $M_+ \times M_-$. This metric encapsulates the problem's geometric signature and induces a symplectic structure via

$w(P, Q) = h(P, U(Q)),$

with $U$ an involutive map switching tangent factors, reminiscent of almost complex structures in Kähler and symplectic geometry. These structures control the regularity, integrability, and signature properties of OT solutions, directly affecting the existence and uniqueness of Monge maps and the properties of Kantorovich measures (McCann, 2012).

2. Monge Maps, Kantorovich Measures, and Regularity via Geometry

Existence, uniqueness, and regularity properties of OT solutions are cast in geometric terms. In the Monge setting, a measurable map $G:M_+ \to M_-$ solves

$\inf_G \int_{M_+} c(x, G(x)) \, d\mu_+(x),$

with regularity (e.g., invertibility or smoothness) determined by the nondegeneracy and signature of the cost Hessian as specified by the cross-difference $\delta$. The Ma–Trudinger–Wang conditions are recast in this geometric language, relating regularity to the curvature and topology of $h$.

In the Kantorovich relaxation, minimization over joint couplings $\gamma$ on $M_+ \times M_-$ yields concentration of $\operatorname{spt}\gamma$ on "spacelike" submanifolds with respect to $h$. The local pseudo-metric structure and the c-monotonicity property directly tie the dimension and rectifiability of the support of optimal $\gamma$ to geometric equations, with bounds derivable from the signature of the Taylor expansion (McCann, 2012).

3. The Fusion Principle: From Cost Geometry to Fused Objects

Geometry-driven optimal transport fusion formalizes the idea that the geometry of the cost controls the structure of the optimal transport and the fused objects thereafter:

• The cross-difference $\delta$ and its Taylor expansion serve as the analytic bridge connecting local infinitesimal geometry to global properties of transport plans and mappings.
• The c-cyclical monotonicity and noncriticality conditions (such as those underpinning the existence of Lipschitz dual potentials) translate into geometric constraints on graphs of optimal mappings.
• The regularity and uniqueness of Monge solutions are directly linked to geometric properties such as the curvature, metric signature, and geodesic convexity induced by $c$.

When considering the "fusion" of mass distributions, models, or other mathematical objects, these geometric facts guarantee that fusion—understood as the natural combination or composition of optimized couplings or maps—respects the underlying geometry encoded by the cost function.

4. Linear Programming Duality and the Geometric Structure of Potentials

The dual formulation of the OT problem,

$$\min_\gamma \iint c(x, y) \, d\gamma(x, y) = \sup_{(u, v) \in \mathrm{Lip}_c} \left\{ \int u(x) d\mu_+(x) + \int v(y) d\mu_-(y) \right\},$$

means that dual optimizers (potentials) $u$ and $v$ are not merely optimization variables but encode geometric information. When differentiable, their gradients are linked to the OT cost through $D_xc(x, G(x))$, making duality itself a geometric phenomenon.

The relationship to differential topology is deepened by the fact that integration of relevant differential forms over loops in $M_+$ yields constraints for the existence of potentials, and the structure of these potentials is governed by $h$ and its curvature.

5. Metric Invariants, Curvature, and Connections to Symplectic/Kähler Geometry

The pseudo-metric $h$ has as many negative as positive eigenvalues in the generic (full-rank) case, so the OT problem naturally sits in the field of indefinite metric geometry. The induced symplectic form $w$ introduces another invariant, connecting OT with Kähler and symplectic geometric settings. For certain cost functions (e.g., the quadratic cost), the graph of an optimal map becomes a zero-mean curvature surface maximizing the $h$-volume among homologous surfaces—reminiscent of volume-minimizing and maximal surfaces in geometric analysis.

The geometric/topological invariance of regularity conditions such as the Ma–Trudinger–Wang condition—here expressed via the cross-difference and its Hessian rather than coordinate-specific derivatives—highlights the role of the intrinsic geometry of the transport cost (McCann, 2012).

6. New Analytical and Heuristic Connections

A notable contribution is the translation of classical coordinate-based regularity conditions into intrinsic differential-geometric statements, yielding:

• A broader context for geometric measure-theoretic methods in OT.
• An explicit link between mean curvature, structural maximality of the transport graph, and the transport cost's geometry.
• Connections to linear programming and functional analysis via the geometry-driven structure of dual potentials.

This suggests the potential for further analytic results on regularity, uniqueness, and dimension bounds to be established or strengthened by focusing on intrinsic properties of the transport cost and the induced pseudo-metric and symplectic structures.

7. Implications and Areas for Further Research

Geometry-driven optimal transport fusion points toward several research directions:

• Using geometric invariants (metric signature, curvature, symplectic structure) to develop new regularity conditions or error estimates in complex OT settings, including those on manifolds with nontrivial topology.
• Employing the fusion of geometric and topological methods—such as those from geometric measure theory or Kähler geometry—to analyze and optimize multi-marginal OT, branched transport, or other generalizations.
• Extending the geometric paradigm to high-dimensional and computational settings, where understanding the emergent "fused" object’s structure is essential for both theoretical insight and practical algorithms.
• Investigating how these geometric structures influence the design and convergence of numerical algorithms for OT problems.

Conclusion

Geometry-driven optimal transport fusion interprets optimal mass transport not as solely an optimization problem but as one with rich differential and topological underpinnings, where the cost function's geometry shapes the regularity, uniqueness, and qualitative features of OT solutions. Through constructs such as the cross-difference, pseudo-Riemannian metrics, and associated symplectic forms, this framework unites classical analysis with modern geometric thinking, providing a foundation for both theoretical advances and robust, geometry-aware transport-based methods in diverse scientific fields (McCann, 2012).