[论文解读] Schrödinger bridge problem via empirical risk minimization

Denis Belomestny, Alexey Naumov+2 more|arXiv (Cornell University)|Feb 9, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

paper proposes learning the Schrödinger bridge from samples by formulating a nonlinear fixed-point problem for a transformed potential g and estimating it via empirical risk minimization, leading to a continuous potential and a sampler for the bridge via stochastic control.

ABSTRACT

We study the Schrödinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schrödinger potentials via Sinkhorn iterations on empirical measures and then construct a time-inhomogeneous drift by differentiating a kernel-smoothed dual solution. In contrast, we propose a learning-theoretic route: we rewrite the Schrödinger system in terms of a single positive transformed potential that satisfies a nonlinear fixed-point equation and estimate this potential by empirical risk minimization over a function class. We establish uniform concentration of the empirical risk around its population counterpart under sub-Gaussian assumptions on the reference kernel and terminal density. We plug the learned potential into a stochastic control representation of the bridge to generate samples. We illustrate performance of the suggested approach with numerical experiments.

研究动机与目标

• Motivate estimating Schrödinger potentials when endpoint marginals are available only through samples.
• Reformulate SBP as a nonlinear fixed-point problem for a transformed potential g.
• Propose ERM to estimate g over a function class, enabling off-sample generalization.
• Establish uniform concentration of the empirical risk under sub-Gaussian assumptions on kernel and terminal density.
• Demonstrate a practical sampler by plugging the learned potential into a stochastic control representation.

提出的方法

• Rewrite SBP with a single transformed potential g that satisfies a nonlinear fixed-point equation g = C[g].
• Define empirical operator ĈN,M[g] by replacing expectations with sample averages.
• Form ERM objective Ŕ̂N,M(g) = (1/M) Σj ℓ(g(Yj), ĈN,M[g](Yj)) with ℓ minimizing at equality (e.g., squared loss).
• Optimize over a hypothesis class 𝒢 of positive functions (e.g., neural networks) via stochastic gradient methods.
• Prove uniform concentration: E[R(ĝN,M)] ≤ infg∈𝒢 R(g) + 2 E[ supg∈𝒢 | Ŕ̂N,M(g) − R(g) | ].
• Show that with Gaussian Q and Hermite expansion, the population fixed point admits a rapidly converging series and provide end-to-end risk bounds.
• Use the learned potential in the stochastic control form to generate bridge samples by drift adjustment (a∇log h with h(t,x) evolution).

Figure 1 : Sample translation from Swiss-Roll to S-Curve and density map for ERM-Bridge at time $t\in[0,0.25,0.5,0.75,1].$

实验结果

研究问题

• RQ1 Can a single transformed potential g capturing the Schrödinger system be learned directly from samples?
• RQ2 What are the statistical guarantees (concentration and approximation) for the ERM estimator of g under sub-Gaussian kernel/terminal density assumptions?
• RQ3 How well does the learned potential enable sampling of Schrödinger bridges via stochastic control representations?
• RQ4 What are the trade-offs between continuous, learned potentials versus discrete Sinkhorn-based potentials in terms of accuracy and generalization?
• RQ5 How does the proposed method perform on synthetic and real-data transport tasks compared to existing baselines?

主要发现

• ERM-Bridge yields a continuous learned potential suitable for drift construction without post-hoc smoothing.
• Under Gaussian reference kernel, the population fixed point g⋆ admits an efficient Hermite function expansion, enabling explicit approximation bounds.
• Uniform concentration of the empirical risk around the population risk is established with near-parametric dependence on sample size (up to polylog factors).
• Numerical experiments show competitive or superior performance to baselines on Swiss roll to S-curve, Gaussian mixture transport under shift, and single-cell interpolation tasks.
• The method produces smoother, continuous potentials and comparable or improved transport quality versus Sinkhorn-based approaches, with favorable training and sampling times.
• The framework accommodates sparse representations and scalable evaluation of potentials and gradients for drift computation.

Figure 3 : Comparison of training and sampling times for ERM-Bridge and SinkhornBridge on the Swiss-Roll to S-Curve translation task. We used 2000 training points for both algorithms.