Adapted Optimal Transport between Filtered Gaussian Processes

Abstract: We continue the study of adapted optimal transport in the discrete-time Gaussian setting. To this end, we introduce a space of filtered Gaussian processes where both the randomness and the flow of information are driven by a Gaussian white noise. On this space, the adapted $2$-Wasserstein distance (${AW}_2$) admits a variational representation as a constrained orthogonal Procrustes problem between Cholesky factors. Furthermore, the resulting quotient space is the ${AW}_2$-completion of the space of Gaussian distributions on the path space. We also characterize explicitly the ${AW}_2$-projections onto the subspaces of Gaussian martingales. Next, we analyze the adapted Brenier coupling -- a multivariate generalization of the Knothe--Rosenblatt coupling that serves as a myopic solution to the adapted transport problem, and compute its transport cost. Utilizing a Gaussian random matrix framework, we investigate the asymptotic behavior of transport costs as the time horizon grows; notably, we establish that the transport costs of all Gaussian bicausal couplings are asymptotically equivalent, whereas the classical Bures--Wasserstein distance is strictly smaller. Finally, we demonstrate that the adapted analogue of Gelbrich's lower bound fails in general, and we identify a sufficient martingale difference condition under which the bound is recovered.

• The paper introduces a formal framework for filtered Gaussian processes and derives an explicit adapted 2-Wasserstein (AW2) formula for measuring transport costs under temporal constraints.
• It presents a variational representation that reformulates AW2 as a constrained orthogonal Procrustes problem, offering actionable insights for sequential modeling.
• The study analyzes asymptotic transport costs and demonstrates the failure of the adapted Gelbrich bound, impacting robust uncertainty quantification in causal settings.

Summary of "Adapted Optimal Transport between Filtered Gaussian Processes" (2604.22159)

Background and Motivation

Adapted optimal transport (AOT) extends classical optimal transport (OT) to stochastic processes, imposing filtration and the causality (temporal information flow) constraints relevant in fields like mathematical finance and sequential modeling. While classical OT ignores temporal structure, AOT demands that couplings cannot "see into the future" (bicausality). Explicit calculations in AOT are rare, but Gaussian processes provide a tractable setting. This work builds on recent progress in adapted Wasserstein distances ($AW_2$) in discrete-time Gaussian settings, introducing a formal framework for filtered Gaussian processes and advancing the theory, including explicit computations, variational representations, and analytic comparison of transport costs.

Core Contributions

Definition and Structure of Filtered Gaussian Processes

The paper formalizes filtered Gaussian processes as processes $X = a + L\epsilon$, where $a$ is a mean vector, $L$ a block lower-triangular matrix (Cholesky factor), and $\epsilon$ a Gaussian white noise. This structure accommodates both degenerate and non-degenerate covariance, and allows flexibility in the filtration (temporal information). Filtered Gaussian processes are richer than mere Gaussian distributions since different Cholesky factors can yield identical covariance but encode different information structures.

Adapted $2$-Wasserstein Distance ($AW_2$) and Variational Representation

A principal result is an explicit formula for $AW_2$ between two filtered Gaussian processes: for $X = a + L\epsilon$, $Y = b + M\epsilon$,

$X = a + L\epsilon$0

where the norms and block structure are detailed in the paper. This cost admits a variational interpretation as a constrained orthogonal Procrustes problem, minimizing $X = a + L\epsilon$1 over block-diagonal orthogonal matrices $X = a + L\epsilon$2, where each block is optimally aligned to maximize nuclear trace terms.

Completion and Geometric Properties

The completion of the space of Gaussian distributions under $X = a + L\epsilon$3 is characterized as the quotient space of filtered processes (modulo zero-distance equivalence), yielding a Polish metric space. The quotient captures the degeneracy inherent in the Cholesky ambiguity: two factors related by blockwise orthogonal matrices have zero pseudo-distance.

Adapted Brenier Coupling and Stepwise Myopic Solutions

An adapted analogue of the Brenier coupling is analyzed. The classical Brenier coupling provides the optimal OT map for Gaussian distributions; its adapted counterpart is a multivariate generalization of the Knothe-Rosenblatt coupling, enforcing step-by-step (myopic) optimality under information constraints. Explicit formulas for its transport cost are provided. Notably, the adapted Brenier cost is not a squared metric, diverging from $X = a + L\epsilon$4 and classical OT.

Asymptotics and Comparison of Transport Costs

Using a random matrix framework ($X = a + L\epsilon$5 i.i.d. Gaussian entries), the authors analytically show that as the time horizon grows ($X = a + L\epsilon$6 with fixed $X = a + L\epsilon$7), all Gaussian bicausal couplings (including adapted Brenier) yield asymptotically equivalent transport costs, scaling as $X = a + L\epsilon$8. However, the classical Bures–Wasserstein distance (physical OT between entire trajectories, no filtration constraints) is strictly smaller, demonstrating a gap imposed by causality.

Adapted Gelbrich Bound Failure and Sufficient Condition

Gelbrich's lower bound for classical $X = a + L\epsilon$9 (the distance between arbitrary measures is at least the OT between their Gaussian projections) does not extend to the adapted setting: $a$0 between processes can be strictly less than between their Gaussian projections. The paper proves this failure in general, but provides a sufficient condition—martingale difference property of the innovations—for the bound to be recovered.

Numerical and Contradictory Results

• Explicit Asymptotics: Transport costs for all Gaussian bicausal couplings asymptotically scale as $a$1, but the Bures–Wasserstein cost is strictly smaller as $a$2.
• Adapted Gelbrich Bound Failure: A constructed counterexample shows that $a$3 between general distributions can be less than $a$4 between their Gaussian counterparts, contradicting classical intuition.

Practical and Theoretical Implications

The results deepen the understanding of adapted Wasserstein distances in sequential Gaussian modeling, with implications for uncertainty quantification and robust decision-making under informational constraints. The variational and Procrustes representations provide algorithmically tractable formulations for sequential loss functions in filtered settings. The failure of the adapted Gelbrich bound implies caution for practitioners employing Gaussian benchmarks for uncertainty in sequential OT. The random matrix asymptotic equivalence suggests robustness of cost estimation across coupling methods in large time horizon problems under Gaussianity.

Future Directions

The theoretical framework invites extensions to non-Gaussian processes, continuous-time models, and analysis of non-linear filtrations. Developing tractable approximations or bounds for general adapted OT problems—beyond the Gaussian or Brenier settings—remains a challenging area. Further study of the geometric structure (tangent space, curvature) of the metric space of filtered processes could yield richer optimization tools for sequential models in finance, control, and machine learning.

Conclusion

The paper develops a comprehensive and computationally explicit theory of adapted optimal transport between filtered Gaussian processes. It advances the mathematical foundation for sequential uncertainty quantification with temporal information, details practical cost formulas and variational principles, and rigorously delineates the regime where classical transport intuition fails under causality constraints. This work sets a new standard for explicit adapted OT analysis in the Gaussian discrete-time framework and provides tools and insights for robust sequential modeling.