Plug-in estimation of Schrödinger bridges

Entropy-regularized optimal transport (OTε) has become a fundamental tool for measuring distributional differences, especially in high-dimensional data. The Sinkhorn algorithm enables efficient computation of OT potentials, which approximate the classical Wasserstein distance with added regularization. Schrödinger bridges (SB), originating from quantum mechanics and large deviation theory, model stochastic paths interpolating between distributions, characterized by stochastic differential equations (SDEs). Recent advances combine OT with SB, enabling dynamic path inference. However, existing methods—like neural SDEs and iterative neural network training—are computationally intensive and lack statistical guarantees. This paper aims to leverage static OT potentials for efficient SB estimation, addressing high computational costs and theoretical gaps.

Estimating the Schrödinger bridge between two unknown distributions in high dimensions remains challenging. Existing approaches require iterative simulation of forward and backward diffusions or neural network training to fit unknown drifts, which are computationally costly and lack rigorous statistical guarantees. The core bottleneck is deriving accurate, scalable estimators for the time-dependent drift functions that define the stochastic paths. Overcoming these limitations is crucial for applications in generative modeling, particle tracking, and stochastic control, where efficiency and statistical robustness are paramount.

The main innovation is exploiting the static entropic OT potentials obtained via Sinkhorn algorithm to construct a plug-in estimator for the time-dependent drift. This bypasses iterative diffusion simulation and neural network training. The approach involves: 1) computing static potentials (́f, ́g) from source and target samples; 2) deriving a closed-form expression for the drift at any time t based on these potentials; 3) simulating paths with discretized SDEs driven by this estimated drift. Theoretically, the estimator converges at rates depending on the target's intrinsic dimension, not ambient dimension, making it effective in high-dimensional settings.

• �� Sample data: draw m samples from source μ and n samples from target ν. • Compute static potentials (́f, ́g) via Sinkhorn algorithm on these samples, with complexity O(mn/ε). • Construct the drift estimator ́b_t(z) using the potentials, involving exponential weights and Gaussian kernels. • Derive the time-dependent drift ́b_t(z) as a closed-form expression involving the potentials. • Simulate paths by discretizing the SDE with drift ́b_t(z), starting from μ. • Analyze the total variation risk bounds, showing convergence rates depending on the target's intrinsic dimension, with robustness to high ambient dimensions.

Experiments include synthetic high-dimensional data, where the estimator’s error decreases at parametric rates with sample size, confirming theoretical predictions. The Föllmer bridge scenario, with a Dirac source and smooth target on a k-dimensional manifold, demonstrates that error bounds depend only on k. Comparisons with neural network-based diffusion models show reduced computational cost and improved statistical guarantees. Hyperparameters like regularization ε, sample size, and discretization steps are tuned to optimize performance. Additional robustness tests validate the method’s effectiveness across different target complexities and noise levels.

The estimator achieves error bounds that scale with the target’s intrinsic dimension k, with sample complexity approximately R^{2(1- au)^{-k-2}\epsilon^{-2}} for fixed error levels. In the Föllmer case, the total variation error diminishes at rates depending solely on k, confirming the method’s efficiency in high dimensions. Theoretical risk bounds show divergence as ε → 0 or τ → 1, but the growth is controlled by intrinsic dimension. Empirical results validate these bounds, demonstrating the estimator’s robustness and scalability.

The method is applicable to high-dimensional data migration, generative modeling, and stochastic control. It enables efficient path sampling and distribution interpolation with minimal assumptions, suitable for large-scale machine learning tasks like image synthesis, protein folding, and particle tracking. The approach requires only source and target samples, making it accessible for real-world applications where explicit density models are unavailable.

The estimator’s performance deteriorates as noise ε approaches zero, limiting near-deterministic regimes. In extremely high-dimensional or irregular target distributions, potential estimation accuracy may decline. Numerical stability depends on Sinkhorn algorithm implementation, which can be sensitive to hyperparameters. Future work should focus on adaptive regularization, potential estimation robustness, and extending to multimodal or non-smooth targets.