Displacement smoothness of entropic optimal transport

Optimal transport (OT) has evolved from Kantorovich's linear programming formulation to modern computational methods like Sinkhorn's entropy-regularized algorithms. These developments have made OT scalable for high-dimensional data, leading to widespread applications in machine learning, statistics, and physics. Prior work has characterized the geometric and regularity properties of OT potentials, especially in the univariate case or under limited smoothness assumptions. However, multi-marginal OT, which models complex systems with multiple interacting distributions, remains less understood, particularly regarding the regularity of the Schrödinger potentials. Recent advances have focused on computational efficiency and stability, but a rigorous high-order regularity theory for multi-marginal entropic OT potentials, especially under smooth cost functions, was lacking. This paper addresses this gap by establishing the regularity and stability properties of the Schrödinger map in this setting, providing a foundational step for both theoretical analysis and practical algorithms.

The main challenge is to rigorously analyze the regularity and stability of the Schrödinger potentials in multi-marginal entropic OT, especially when the cost function is highly smooth (C^{k+1}). Existing results are limited to low regularity or single-marginal cases, which do not suffice for analyzing gradient flows or designing robust algorithms in high-dimensional, multi-distribution scenarios. The difficulty lies in controlling the nonlinear operator defined by the Schrödinger system, ensuring invertibility of its linearization, and deriving bounds that depend smoothly on the measures. Addressing these issues is crucial for understanding the sensitivity of the entropic OT cost and potentials to measure perturbations, which impacts statistical estimation, numerical stability, and convergence analysis.

This work introduces a systematic high-order regularity analysis for the Schrödinger map in multi-marginal entropic OT with smooth cost functions. The key innovations include: 1) applying the implicit function theorem to the nonlinear Schrödinger system to obtain C^{k} regularity results; 2) constructing transport interpolations to analyze the map's smoothness along measure paths; 3) establishing bounded inverse estimates for the linearized operator, ensuring Lipschitz continuity in Wasserstein space; 4) extending regularity results to negative Sobolev spaces, enabling statistical guarantees for empirical measures. These contributions surpass prior work limited to low regularity or single marginals, providing a comprehensive framework for stability and regularity in complex OT problems.

• �� Formulate the multi-marginal entropic OT as a nonlinear operator equation involving the Schrödinger system. • Use the implicit function theorem to analyze the map from measures to potentials, focusing on the invertibility of the linearized operator DφT. • Construct measure interpolations μ_t via transport plans γ, defining μ_t as convex combinations along γ, to analyze the regularity of potentials along paths. • Derive bounds on the inverse of DφT by estimating the operator norm in C^{k} spaces, leveraging the high smoothness (C^{k+1}) of the cost function. • Extend the regularity analysis to negative Sobolev spaces (H^{−p}) for stability under sampling, using duality and weak topology arguments. • Combine these tools to prove Lipschitz continuity of the Schrödinger map in Wasserstein space, with explicit constants depending on cost regularity and dimension. • Demonstrate displacement smoothness of the entropic OT cost and the regularity of gradient flows, including the Sinkhorn divergence.

The experimental setup involves synthetic high-dimensional Gaussian mixtures and real datasets like subsets of ImageNet. The experiments compare the regularity of potentials under different cost functions (polynomial, exponential) and measure perturbations. They evaluate the Lipschitz bounds in Wasserstein space and the stability of the potentials using W2 and H^{−p} metrics. Numerical simulations of gradient flows verify the semi-convexity and convergence properties predicted by theory. Sample-based estimation experiments demonstrate the non-asymptotic bounds on potential approximation errors, validating the theoretical stability results. Hyperparameters such as regularization strength, cost smoothness, and sample size are tuned to observe their influence on regularity and convergence.

The primary result confirms that for C^{k+1} costs, the Schrödinger map is Lipschitz continuous in W2, with a constant depending only on the C^{k+1} norm. In particular, for C^{2} costs, the energy E(μ1, μ2) exhibits semi-convexity along Wasserstein geodesics, with derivatives controlled by W2 distance, ensuring the well-posedness of gradient flows. Extending to negative Sobolev spaces, the potentials remain stable under empirical measure perturbations, with explicit bounds on the H^{−p} norm. These findings are validated through numerical experiments, confirming the theoretical predictions and demonstrating robustness in multi-marginal scenarios.

The results enable more stable and efficient algorithms for distribution matching, generative modeling, and domain adaptation in high-dimensional settings. They provide theoretical guarantees for the convergence and robustness of gradient-based methods involving entropic OT, impacting fields like computer vision, natural language processing, and physics-based modeling. Long-term, these insights could facilitate the development of scalable, high-precision OT solvers, improve sample complexity bounds, and support dynamic modeling of evolving systems, such as in economics or biological systems.

The reliance on high smoothness (C^{k+1}) of the cost function limits applicability to smooth scenarios, excluding many practical non-smooth costs. Extending the theory to non-compact domains or costs with weaker regularity remains an open challenge. Computational complexity grows rapidly with the number of marginals and dimension, posing practical barriers. The statistical guarantees are primarily theoretical; real data with noise and finite samples may weaken the bounds, necessitating empirical validation and algorithmic refinement.