Wasserstein Distance Stability

• Wasserstein Distance Stability is the study of the robustness of the Wasserstein metric under numerical, distributional, and algorithmic perturbations.
• It examines challenges like underflow in Sinkhorn iterations and bias in entropic regularizations, introducing methods such as IPOT for enhanced reliability.
• The analysis further highlights translation-invariant metrics and quantitative stability for OT potentials, maps, and barycenters in various applied contexts.

The Wasserstein distance is a foundational metric of probability distributions with extensive applications across optimal transport, statistics, machine learning, and geometric analysis. Understanding the stability properties of the Wasserstein distance and related functionals is central to both theory and applications, particularly when quantifying the reliability of computations, measuring robustness to perturbations, or controlling empirical and algorithmic behavior under data shifts.

1. Numerical Stability in Entropy-Regularized Optimal Transport

Entropy-regularized optimal transport, commonly computed via the Sinkhorn algorithm, introduces a trade-off between computational tractability and numerical stability. The regularized $p$-Wasserstein distance is defined, for discrete measures $\mu, \nu$ over finite supports, as

$$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$

where $C$ is the cost matrix, $U(\mu, \nu)$ is the set of couplings with prescribed marginals, and $\epsilon > 0$ is the entropic regularization parameter.

The Sinkhorn iteration alternates scaling left and right via

$$a \leftarrow \mu / (G b), \quad b \leftarrow \nu / (G^T a), \quad \text{where} \quad G_{ij} = \exp(-C_{ij}/\epsilon)$$

and $$T = \operatorname{diag}(a) G \operatorname{diag}(b)$$. Two antagonistic numerical phenomena arise:

• Underflow as $\epsilon \to 0$: $G_{ij} \to 0$, so floating-point underflow is severe, even for log-domain implementations. Furthermore, the Sinkhorn contraction rate $\mu, \nu$0, leading to extremely slow convergence and loss of numerical precision.
• Bias as $\mu, \nu$1: The entropic term dominates, yielding a dense, blurred transport plan, incurring an $\mu, \nu$2 bias in the Wasserstein cost and the "shrinkage effect" in generative modeling.

The IPOT (Inexact Proximal Point Optimal Transport) algorithm circumvents this by a sequence of Bregman-proximal steps, never exposing $\mu, \nu$3 with vanishing $\mu, \nu$4 to exponentiation. Each update is

$\mu, \nu$5

where $\mu, \nu$6 is the Kullback–Leibler divergence. This approach guarantees convergence to the exact unregularized solution, eliding underflow/blurring, with cost per iteration similar to a moderately regularized Sinkhorn. IPOT achieves robust regularization and linear convergence under classical assumptions, with empirical tests showing reliable recovery of sparse plans and superior generative-model behavior—avoiding mean-shrinkage and mode collapse—relative to standard entropic regularization methods (Xie et al., 2018).

2. Distributional Stability and Translation-Invariant Wasserstein Metrics

Distributional shifts, especially pure translations, can produce spurious Wasserstein effects which are undesirable in settings involving data with unobservable or irrelevant spatial alignment. The relative-translation-invariant Wasserstein distance $\mu, \nu$7 on the quotient space $\mu, \nu$8 is defined via

$\mu, \nu$9

where $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$0 is the pushforward under translation by $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$1. This metric is

• A genuine metric on equivalence classes under translation,
• Fully translation-invariant: $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$2 for all $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$3,
• Decomposable ($$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$4): $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$5, i.e., a Pythagorean "bias–variance" split.

Applications include:

• Empirical evidence that $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$6 yields more robust classification under spatial translation, e.g., on MNIST or radar-based tasks;
• Efficient computation via a variant of Sinkhorn alternating between translation and scaling steps, with controlled sensitivity and time complexity;
• Theoretical guarantees and translation-invariant guarantees on shift-related distributional perturbations (Wang et al., 2024).

3. Quantitative Stability of OT Potentials, Maps, and Projections

Stability quantifies to what extent optimal transport plans, maps, and associated functionals respond to perturbations in the input measures.

• Kantorovich potentials: $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$7 with explicit exponents: $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$8 for $$W_\epsilon(\mu, \nu) = \min_{T \in U(\mu, \nu)} \langle T, C \rangle + \epsilon \sum_{i,j} T_{ij} \ln T_{ij}$$9, $C$0 (Mischler et al., 2024).
• Brenier maps (for $C$1): $C$2.
• Wasserstein projections (Shadow mapping): For product spaces and $C$3, shadow projections depend (bi-)Hölder-continuously on variations in marginals and reference measures, with explicit exponents controlled by power cost and regularity, enabling sample complexity rates for empirical projections (Kim, 20 Apr 2026).

4. Functional and Geometric Stability: Barycenters, Sliced, and Max-Sliced Wasserstein

• Wasserstein barycenters: On Alexandrov spaces with curvature bounds, the barycenter map is globally Hölder-continuous, with an explicit exponent $C$4 summarizing the rate at which barycenters respond to perturbations in underlying law—empirically, for $C$5 input measures (or batch) with $C$6 perturbation, the barycenter varies as $C$7 (Han et al., 25 May 2026).
• Sliced Wasserstein deficit: The difference $C$8 vanishes only if the Brenier map is homothetic affine. Rigidity and explicit spectral gap ("sliced Poincaré–Korn" constant) conditions quantify the gap between full and sliced Wasserstein, with sharp constants for Gaussian reference and explicit counterexamples showing the necessity of new stability criteria outside standard curvature or Poincaré controls (Han, 25 May 2026).
• $C$9-plane property: For Radon measures with moment bounds, the $U(\mu, \nu)$0-plane transform and max-sliced Wasserstein $U(\mu, \nu)$1 both admit two-sided Hölder comparisons to $U(\mu, \nu)$2, ensuring stability of geometric data under measurement projections (Terzioglu et al., 1 May 2026).

5. Functional and Evolutionary Stability: Flows, PDEs, and Dynamics

• Gradient flows for Sliced Wasserstein: The $U(\mu, \nu)$3 energy yields non-convex dynamics; yet the only stable Lagrangian critical points are atomless measures satisfying a barycentric neutrality. Segment-supported measures are always unstable; discrete gradient descent with step size scaling as $U(\mu, \nu)$4 (number of particles) provides robust convergence and de facto avoids degenerate solutions, with theoretical guarantees and numerical confirmation (Vauthier et al., 10 Feb 2025).
• Porous medium and kinetic Vlasov–Poisson equations: For nonlinear PDEs, Wasserstein stability bounds capture propagation of uncertainties and long-time behavior. For porous medium flows on manifolds, explicit bounds are proved depending on Ricci curvature and Sobolev inequalities, generalizing prior contraction results to negative curvature, with sharpness in the hyperbolic regime (Ponti et al., 2019). In kinetic theory, a fully explicit Osgood-type integral bound is given for kinetic Wasserstein stability with Yudovich macroscopic densities, unifying prior results under a general $U(\mu, \nu)$5-Wasserstein framework (Junné et al., 10 Mar 2025).

6. Invariant and Adapted Stability Notions

• Temporal model risk: In stochastic process-based finance, model proximity in classical $U(\mu, \nu)$6 does not imply closeness of hedging/risk metrics. The "adapted Wasserstein" $U(\mu, \nu)$7 distances incorporate filtration/concretized time information, yielding sharp Lipschitz continuity of risk measures and maintaining stability under model perturbations, as opposed to classical $U(\mu, \nu)$8 which can allow discontinuous statistics (Backhoff-Veraguas et al., 2019).
• Ergodicity and cutoff behavior: For OU and Jacobi chain dynamics, sharp cutoff phenomena are demonstrated in $U(\mu, \nu)$9, with explicit scaling results ("abrupt thermalization") and precise window cutoffs leveraging shift-linearity (Barrera et al., 2023).

7. Stability in Topological and Geometric Data Analysis

• Persistence diagrams and landscapes: Stability in the $\epsilon > 0$0-Wasserstein sense, for both barcodes and persistence landscapes, refines classical bottleneck (supremum) stability. Using natural metrics (e.g., the rank-invariant area for interval modules), it is shown that barcodes and landscapes vary continuously and quantitatively with cellwise perturbations, enabling robust and interpretable topological summaries in data pipelines (Skraba et al., 2020, Zhao et al., 25 Sep 2025).

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The multifaceted stability theory of Wasserstein distances encompasses numerical, geometric, functional, algorithmic, and statistical robustness. Key advances include the elimination of instability in Sinkhorn/IPOT schemes, robust metrics for distributional shift, quantitative perturbation controls (Hölder, Lipschitz, convexity constants), and structurally aware metrics for dynamic and topological problems. These collectively establish the Wasserstein framework as not only a metric space theory but also a stable, reliable foundation for modern applied mathematics and machine learning.