Optimal Transport Gaussian Process

• OT-GP is a framework that combines optimal transport and Gaussian processes to compute closed-form distances and transport maps between Gaussian measures.
• It employs quadratic cost functions and spectral techniques to develop tractable numerical schemes for kernel regression, distributional inference, and model alignment.
• The approach supports diverse applications including surrogate kernel modeling, scalable generative learning with Sinkhorn divergences, and efficient barycenter computation.

Optimal Transport Gaussian Process (OT-GP) refers to a class of models and metrics that combine optimal transport (OT) theory with Gaussian process (GP) frameworks to define distances, kernels, or generative objectives between distributions, random fields, or Gaussian processes. These constructions admit closed-form solutions for the cost of transporting one GP to another under quadratic metrics, provide tractable numerical schemes, and support a variety of learning and inference applications, such as spectral estimation, kernel regression, distributional inference, and model alignment.

1. Foundations: Optimal Transport and Gaussian Processes

Optimal transport addresses the problem of defining the minimal cost to move probability mass from one distribution to another, typically under a cost function such as squared Euclidean distance. In the context of Gaussian processes—viewed as Gaussian measures on infinite-dimensional Hilbert spaces—the OT cost takes a particularly tractable form when both marginal processes or fields are jointly Gaussian. The prototypical cost function is quadratic, leading to the 2-Wasserstein or Bures–Wasserstein geometry.

The OT distance $W_2^2$ between centered Gaussian measures $N(0, A)$ and $N(0, B)$ with covariance operators $A$ and $B$ is

$$W_2^2(N(0,A),N(0,B)) = \operatorname{Tr}A + \operatorname{Tr}B - 2\operatorname{Tr}(A^{1/2}BA^{1/2})^{1/2}$$

which remains valid for trace-class, infinite-dimensional operators and is known as the Bures–Wasserstein metric (Yun et al., 25 Dec 2025).

2. Spectral-Domain OT and Hellinger Distances for Stationary Fields

For zero-mean, multivariate, stationary Gaussian random fields $x_t, y_t$ on $\mathbb{Z}^d$, the OT problem can be formulated in terms of their power spectral densities (PSDs) $\Phi_x(e^{j\omega})$, $\Phi_y(e^{j\omega})$. The optimal transport cost for a quadratic filter $N(0, A)$0, corresponding to frequency-domain weight $N(0, A)$1, is given by

$N(0, A)$2

where $N(0, A)$3 and $N(0, A)$4 denote the PSDs, and $N(0, A)$5 is a weighted matrix Hellinger distance. When $N(0, A)$6, this reduces to the classical (unweighted) Hellinger distance between matrix-valued spectra (Zorzi, 2021, Zorzi, 2020).

This metric admits a geodesic between $N(0, A)$7 and $N(0, A)$8 in the space of PSDs:

$N(0, A)$9

where $N(0, B)$0 is a unitary alignment factor. The geodesic induces a smooth interpolation between the covariance kernels in the spatial domain via the (inverse) multidimensional Fourier transform.

3. Operator-Theoretic and Hilbert Space Formulations

In infinite-dimensional, possibly degenerate settings, classical duality-based approaches to OT are inapplicable, as the optimal Kantorovich potential is infinite almost everywhere. An operator-theoretic framework leverages Green's operators and the Douglas factorization lemma to construct explicit Monge maps $N(0, B)$1 between Gaussian processes:

$N(0, B)$2

when $N(0, B)$3 is invertible, with all optimal couplings remaining Gaussian and characterized entirely by covariance operators. For degenerate or infinite-dimensional covariance operators, existence and uniqueness of OT maps and couplings are governed by the positivity of the Hilbertian Schur complement $N(0, B)$4, the ranges and kernels of operators $N(0, B)$5, $N(0, B)$6, and the action of partial isometries (Yun et al., 25 Dec 2025).

This framework extends to the construction of barycenters in Wasserstein space by optimizing the Hilbert-Schmidt norm of averaged Green's operators, leading to explicit algorithms for Wasserstein Fréchet means (Yun et al., 25 Dec 2025).

4. Entropic OT and Sinkhorn Divergence Between Gaussian Processes

The entropic regularization of OT (Sinkhorn divergence) facilitates computation and statistical estimation in high or infinite dimensions. For Gaussian processes $N(0, B)$7, $N(0, B)$8 on domain $N(0, B)$9, the 2-Sinkhorn divergence is given by

$A$0

where

$A$1

with $A$2 (Mallasto, 2021). The convergence of the empirical Sinkhorn divergence estimated from finite-dimensional GP marginals is almost sure with rate $A$3 (Mallasto, 2021).

5. OT-based Kernels and Surrogate Models

The integration of OT metrics into kernel methods enables non-Euclidean similarity in structured data. In neural architecture search, a negative-definite tree-Wasserstein distance $A$4 is used as a similarity measure between network architectures, giving rise to the OT-GP kernel:

$A$5

where $A$6 is constructed as a convex combination of tree-based Wasserstein distances on bag-of-operations, indegree, and outdegree distributions, ensuring the resulting kernel is positive-definite (Nguyen et al., 2020).

The resulting surrogate GP models are trained by maximizing the marginal likelihood, with analytic gradients available with respect to all hyperparameters. Empirically, such OT-GP surrogates surpass classical kernels in sample efficiency and predictive accuracy on neural architecture search benchmarks (Nguyen et al., 2020).

6. OT-Driven Generative Learning with Latent Gaussian Processes

In generative modeling of temporal or structured data, OT objectives provide alignment when direct correspondences between samples are unavailable. In latent GP-OT (LGP-OT) models for scRNA-seq time course analysis, a heteroscedastic GP prior parametrizes latent trajectories, and training is driven by an entropic OT loss that matches the generated and observed distributions at each time slice:

$A$7

where $A$8 encodes pairwise costs and $A$9 is the entropic regularization. This approach accommodates cell-type and time asynchrony, does not rely on explicit likelihoods $B$0, and is highly scalable due to Hilbert-space GP approximation and the use of Sinkhorn iterations (Balik et al., 20 May 2026).

7. Adapted and Causal OT Distances for Discrete-Time Gaussian Processes

An adapted (bicausal) 2-Wasserstein distance $B$1 for discrete-time Gaussian processes is defined by minimizing over bicausal couplings. For $B$2 and $B$3, with Cholesky factors $B$4, the adapted Bures-Wasserstein distance is

$B$5

with fully explicit bicausal couplings and closed-form computation at cubic complexity in state dimension (Gunasingam et al., 2024).

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• Operator-theoretic OT for Gaussian processes and barycenters: (Yun et al., 25 Dec 2025)
• Matrix Hellinger distances and spectral geodesics for stationary random fields: (Zorzi, 2021, Zorzi, 2020)
• Sinkhorn divergence for Gaussian processes: (Mallasto, 2021)
• Tree-Wasserstein OT kernels in GP surrogates: (Nguyen et al., 2020)
• Hilbert-Space LGP-OT model for generative learning: (Balik et al., 20 May 2026)
• Adapted OT for causal or discrete-time Gaussian processes: (Gunasingam et al., 2024)
• Sinkhorn-embedding OT kernels between probability distributions: (Bachoc et al., 2022)
• Closed-form and IGW extensions for multimarginal and barycenter problems: (Dandapanthula et al., 3 Dec 2025)

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OT-GP covers a spectrum of methodologies from explicit spectral geometry to scalable generative modeling, unified by the principle that optimal transport endows spaces of Gaussian processes and their practical applications with a tractable, information-rich geometry.