Marginal flows of non-entropic weak Schrödinger bridges
Abstract
This paper introduces a dynamic formulation of divergence-regularized optimal transport with weak targets on the path space. In our formulation, the classical relative entropy penalty is replaced by a general convex divergence, and terminal constraints are imposed in a weak sense. We establish well-posedness and a convex dual formulation of the problem, together with explicit structural characterizations of primal and dual optimizers. Specifically, the optimal path measure is shown to admit an explicit density relative to a reference diffusion, generalizing the classical Schrödinger system. For the pure Schrödinger case, i.e., when the transport cost is zero, we further characterize the flow of time marginals of the optimal bridge, recovering known results in the entropic setting and providing new descriptions for non-entropic divergences including the chi-divergence.
Summary
This paper addresses the problem of divergence-regularized optimal transport in a dynamic setting with weak terminal constraints. The authors extend the classical entropic regularization of optimal transport to a more general framework using convex divergences instead of relative entropy. They formulate a dynamic optimal transport problem on path space, where the discrepancy between probability measures is quantified by a general convex divergence, and the terminal constraint is imposed in a weak sense using a weak optimal transport cost. The approach involves establishing well-posedness and a convex dual formulation of the problem. The authors derive explicit structural characterizations of the primal and dual optimizers. Specifically, they show that the optimal path measure admits an explicit density relative to a reference diffusion, generalizing the classical Schrödinger system. In the pure Schrödinger case (zero transport cost), they further characterize the flow of time marginals of the optimal bridge, recovering known results for entropic regularization and providing new descriptions for non-entropic divergences, particularly the chi-squared divergence. This work generalizes existing results in the entropic case and provides novel insights into the structure of optimal transport plans under general divergence regularization. This research matters to the field because it addresses limitations of entropic regularization, such as overspreading and numerical instability, by exploring alternative divergence penalties. The dynamic formulation and the characterization of optimal plans and marginal flows provide a theoretical foundation for developing new algorithms and applications in areas like machine learning, statistics, and data science, where optimal transport is increasingly used. The weak target formulation also provides added flexibility in modeling broader relationships between distributions.
Key Insights
- •The paper introduces a dynamic formulation of divergence-regularized optimal transport with weak targets, extending beyond the traditional entropic regularization.
- •It establishes a convex dual formulation of the problem, providing a way to compute the optimal transport cost and potentials.
- •The optimal path measure is shown to have an explicit density relative to a reference diffusion, generalizing the classical Schrödinger system. The density is proportional to `dμ_0/dν_0(X_0) * ∂_x ℓ^*(-Q_c φ*(X_T) - C(X) - ψ*(X_0))`, where `ℓ^*` is the convex conjugate of the divergence function, `φ*` is the dual optimizer, `Q_c` is a weak cost function and `ψ*` is a measurable function.
- •For the pure Schrödinger case, the flow of time marginals of the optimal bridge is characterized, recovering known results in the entropic setting and providing new descriptions for non-entropic divergences like the χ²-divergence. The flow of marginals equation is: `∂_x log(Q*_t(x)) = E_Q*[α*_t + ←α_(T-t) ◦ ←T|X_t = x] - ∂_x U(x)`.
- •The paper provides verification theorems (Theorem 3.8) for checking the optimality of solutions, showing that if a measure Q takes the form specified in the theorem, then it is primal optimal and the associated function φ is dual optimal. The functions φ and ψ satisfy a Schrödinger system (3.3).
- •The framework allows for weak terminal constraints, providing flexibility in situations where the target distribution is not precisely known. This is achieved using weak optimal transport costs.
- •A key limitation is the need for superadditivity of the divergence operator for the characterization of optimal plans when the initial measures of the reference and admissible measures do not coincide (μ_0 != ν_0).
Practical Implications
- •The results can be used to develop new algorithms for computing divergence-regularized optimal transport, potentially overcoming the limitations of entropic regularization methods.
- •The characterization of optimal plans and marginal flows can inform the design of machine learning models that rely on optimal transport, such as generative models and domain adaptation techniques.
- •The framework can be applied in various fields, including image processing, manifold learning, and statistical inference, where optimal transport is used to compare and manipulate probability distributions.
- •The weak target formulation opens up new possibilities for modeling uncertainty and robustness in optimal transport applications.
- •Future research directions include developing efficient numerical methods for solving the dynamic divergence-regularized optimal transport problem, exploring the properties of different divergence penalties, and applying the framework to real-world datasets.