Displacement Interpolation in Optimal Transport

• Displacement interpolation is a nonlinear method based on optimal mass transport that constructs geodesics between probability measures while preserving mass and geometric features.
• It utilizes techniques such as monotone rearrangement, Radon slicing, and entropic regularization to achieve efficient and stable interpolation in both one and multi-dimensional spaces.
• Applications span model reduction, image morphing, distributed consensus, and statistical testing, highlighting its broad impact in scientific computing and data analysis.

Displacement interpolation denotes a class of nonlinear interpolation schemes in measure and function spaces, rooted in the theory of optimal mass transport (OMT), wherein interpolated states are realized via geodesics in the Wasserstein metric. The canonical form, McCann’s geodesic, traces the evolution between two probability densities or measures by transporting mass along straight-line trajectories according to an optimal plan. This approach preserves mass and geometric features, sharply contrasting with linear (barycentric) interpolation, and underlies advanced methods for model reduction, image morphing, distributed consensus, and more.

1. Mathematical Foundations: Optimal Transport and McCann’s Geodesic

Let $μ_0$, $μ_1$ be probability measures on $\mathbb{R}^n$ with finite second moments. The Monge–Kantorovich optimal transport problem seeks a coupling $π \in Π(μ_0,μ_1)$ minimizing the transport cost $C(π) = \int c(x,y)\, dπ(x,y)$ for a given cost function, typically $c(x,y)=\|x-y\|^2$. For quadratic cost, the minimum $W_2^2(μ_0,μ_1)$ defines the squared 2-Wasserstein distance. When $μ_0$ is absolutely continuous, the optimal plan is induced by a map $T$ pushing $μ_0$ to $μ_1$.

Displacement interpolation, introduced by McCann, constructs a path of intermediate measures: $$μ_t = (\,(1-t)\,\mathrm{id} + t\,T\,)_\# μ_0, \quad t\in[0,1],$$ where the push-forward describes the evolution of mass along uniquely defined geodesics in Wasserstein space. For each $t$, $μ_t$ is itself a minimizer of the barycentric cost: $$μ_t = \arg\min_ρ \big[(1-t)W_2^2(μ_0,ρ) + t\,W_2^2(ρ,μ_1) \big].$$ This construction extends the notion of geodesics in metric spaces to probability measures, yielding constant-speed paths with respect to $W_2$ (Chen et al., 2015).

2. Algorithmic and Computational Aspects

2.1 One-Dimensional Monotone Rearrangement

In 1D, the unique optimal transport map $T$ is given via monotone rearrangement: $T(x) = F_1^{-1}(F_0(x)),$ where $F_0$, $F_1$ are the cumulative distribution functions (CDFs) of $μ_0$, $μ_1$. The displacement interpolant then tracks quantiles: $F_t^{-1}(u) = (1-t)\,F_0^{-1}(u) + t\,F_1^{-1}(u).$ The density $u_t$ at time $t$ is recovered as the derivative of $F_t$. This structure guarantees mass preservation and sharply preserves discontinuities or interfaces, outperforming pointwise linear interpolation in advection/transport-dominated regimes (Rim et al., 2017).

2.2 Multidimensional Displacement Interpolation

Computational complexity rises in higher dimensions. Direct Monge–Ampère solvers are feasible for small data, but for larger problems or in practice, spectral or sliced approaches are used.

A widely adopted approach for multidimensional fields is to employ the Radon transform: Slicing the function along many 1D projections, performing quantile-based interpolation on each, and inverting the transform to reconstruct the field. This "slice-by-slice" procedure preserves geometric features and is robust for reconstructing phenomena such as moving shocks or interfaces in hyperbolic PDEs (Rim et al., 2017).

2.3 Schrödinger Bridges and Entropic Interpolation

Regularizing OMT with an entropy term yields the Schrödinger bridge problem. The associated entropic interpolation provides a stochastic smoothing effect, but converges weakly to displacement interpolation as the regularization parameter $\epsilon \to 0$. The solution involves solving a Schrödinger system via alternating fixed-point updates, underpinned by contraction in the Hilbert metric for stability and computational efficiency. This technique enables robust displacement interpolation even for large datasets, and can be implemented efficiently using log-domain Sinkhorn-like iterations (Chen et al., 2015).

Interpolation Scheme	Dimensionality Support	Key Computational Feature
Monotone rearrangement	1D	Exact, quantile-based, $O(N)$ cost
Radon slice method	$d\geq1$	Slice-wise 1D interpolation, back-proj.
Entropic/Schrödinger	$d\geq1$	Sinkhorn-like, log-domain, contractive

3. Geometric, Variational, and Dynamical Structure

Displacement interpolation is the unique constant-speed geodesic for Wasserstein distance, and thus forms the geometric backbone of metric measure space theory. The dynamic Benamou–Brenier formulation links displacement interpolation to the solution of a continuity equation with kinetic-energy minimization.

The entropy functional is convex along displacement interpolants if and only if the underlying Riemannian manifold has nonnegative Ricci curvature. The Hamiltonian viewpoint recasts displacement interpolants as projections of Hamiltonian flows, generalizing to Finsler geometry and Ricci flows. This structural connection is crucial in geometric analysis, information geometry, and the theory of gradient flows in measure spaces (Lee, 2012).

4. Applications in Model Reduction, Data Augmentation, and Sampling

4.1 Model Order Reduction for PDEs

Displacement interpolation has enabled reduced-order modeling (ROM) for hyperbolic conservation laws and transport-dominated dynamical systems—regimes where linear superposition and classical POD fail due to nonlinearity and non-smoothness of solution manifolds. By interpolating in transport-map space (or equivalently, interpolating CDF inverses), local low-dimensional manifolds are constructed capturing the dominant transport features. This approach yields ROMs with orders-of-magnitude speedup and precision in capturing traveling fronts, shocks, or sharp interfaces, and directly enables robust uncertainty quantification workflows (Rim et al., 2018, Khamlich et al., 2024, Cucchiara et al., 2023).

Synthetic snapshots derived from displacement interpolation (between observed "checkpoints") can be used for data augmentation, enabling accurate continuous-time emulators and enhanced coverage of solution space for regression-based residual correction (Khamlich et al., 2024).

4.2 Machine Learning and Neural Transport

In generative modeling and OT map learning, utilizing the entire trajectory of displacement interpolation (rather than static endpoint maps) leads to more stable adversarial optimization and superior approximation of transport maps. Neural architectures regularized via the Hamilton–Jacobi–Bellman equation for the dual potentials enforce pathwise consistency and smoothness, outperforming classical approaches in high-dimensional image-to-image translation tasks (Choi et al., 2024).

4.3 Distributed Consensus and Wasserstein Barycenters

Stochastic, distributed algorithms leverage pairwise displacement interpolation to reach Wasserstein barycenter consensus. Each agent updates its state by displacement interpolation toward its neighbor, yielding almost sure convergence to the barycenter—with closed-form updates in the Gaussian setting and clear connections to the DeGroot model of opinion dynamics (Cisneros-Velarde et al., 2020).

4.4 Statistical Detection and Distribution Shift Analysis

Displacement interpolations provide a parametrized family of alternatives (interpolated between null and alternative distributions) for weak distribution shift modeling. Statistical testing (e.g., based on the Wasserstein distance) achieves sharp non-asymptotic detection thresholds, and is analytically tractable in one dimension via quantile representations of interpolated measures (Hur et al., 2023).

5. Generalizations, Extensions, and Domain-Specific Variants

• Nonprobability and Signed Data: Extension to non-probability measures and signed functions is achieved by splitting into signed components and carrying out "piecewise" displacement interpolation (Rim et al., 2018).
• Boundary- and Feature-aware Interpolation: In high-dimensional fields (e.g., CFD or medical imaging), boundary-aware and feature-centric displacement interpolation schemes combine OT-based registration with parametric regression over coherent structure coordinates (e.g., shock fronts, vortices) (Cucchiara et al., 2023, Assis et al., 19 Aug 2025).
• Phase-Space and Quantum Systems: The bifractional displacement operator framework generalizes canonical phase-space displacement to provide a continuum between variational representations (Wigner, Weyl, Husimi), relevant for quantum optics and signal processing (Agyo et al., 2014).
• Rigid-Body Kinematics: Specialized displacement interpolation in the space of motions (SE(3)), such as vertical Darboux and cubic circular motions, enables low-degree, rational interpolation of rigid poses for robotics and animation (Schröcker, 2017).

6. Numerical Implementation, Stability, and Practical Considerations

Common computational steps in displacement interpolation comprise:

• Solving the discrete OT problem, typically via the Sinkhorn algorithm with entropic regularization (Chen et al., 2015).
• Computing quantile functions or pseudo-inverses for 1D cases (Rim et al., 2017).
• Constructing intermediate snapshots via push-forward of mass along the transport plan.
• Using log-domain arithmetic and contraction properties in the Hilbert metric for stability in entropic schemes (Chen et al., 2015).
• For high-dimensional data, employing registration of feature-point clouds and boundary-aware mappings with regularization to preserve geometric fidelity (Cucchiara et al., 2023).

Numerical complexity and stability are dictated by the discretization (grid size $N$), the sparsity and structure of the kernel (Q), and the regularization (entropic parameter $\epsilon$). For sparse or FFT-convolutional Q, computational cost can be substantially reduced.

7. Illustrative Examples and Empirical Performance

• One- and Two-dimensional Examples: Benchmark studies in one-dimensional density interpolation demonstrate diffusion-to-transport transitions as $\epsilon$ is decreased. In 2D, displacement interpolation yields sharp, mass-preserving image morphs with physically plausible flow patterns.
• Model Reduction for Parametric Burgers’ Equation: Displacement-interpolation-based reduced models achieve mean relative errors below $0.5\%$, outperforming linear reduced bases, especially for moving shocks (Rim et al., 2018, Khamlich et al., 2024).
• Brain Biomechanical Interpolation: Deep biomechanical guidance corrects for unphysical artifacts inherent in thin-plate spline or RBF interpolation, reducing MSE by half compared to classical schemes in large-scale brain shift registration (Assis et al., 19 Aug 2025).
• Statistical Testing: Tests designed using displacement interpolation have exact phase transitions for detectability under weak alternatives, matching empirical distributions in socioeconomic and scientific reproducibility studies (Hur et al., 2023).

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Displacement interpolation, as defined and extended in the optimal transport literature, provides a geometrically and physically grounded technique for interpolating between distributions, fields, functions, and states across a variety of disciplines. Its synthesis of deterministic, entropic, and stochastic formulations yields robustness, interpretability, and computational tractability, making it a central tool in contemporary applied mathematics, data science, and scientific computing (Chen et al., 2015, Rim et al., 2017, Rim et al., 2018, Cucchiara et al., 2023, Choi et al., 2024).