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A kernel-based method for Schrödinger bridges

Yumiharu Nakano

TL;DR

The paper develops a kernel-based framework for Schrödinger bridges by embedding probability measures into an RKHS and penalizing terminal-marginal constraints via the maximum mean discrepancy. It shows that the Schrödinger problem is equivalent to a McKean–Vlasov stochastic control problem with a kernel-based constraint, and proves that epsilon-optimal controls converge to the unique Schrödinger solution. A neural SDE–based algorithm is introduced to solve the resulting MKV control problem, and the approach is validated on 1D and 2D bridging/interpolation tasks. By unifying entropic regularization, kernel embeddings, and neural stochastic dynamics, the work offers a scalable method for bridging arbitrary initial and terminal marginals in diffusion processes.

Abstract

We characterize the Schrödinger bridge problems by a family of Mckean-Vlasov stochastic control problems with no terminal time distribution constraint. In doing so, we use the theory of Hilbert space embeddings of probability measures and then describe the constraint as penalty terms defined by the maximum mean discrepancy in the control problems. A sequence of the probability laws of the state processes resulting from $ε$-optimal controls converges to a unique solution of the Schrödinger's problem under mild conditions on given initial and terminal time distributions and an underlying diffusion process. We propose a neural SDE based deep learning algorithm for the Mckean-Vlasov stochastic control problems. Several numerical experiments validate our methods.

A kernel-based method for Schrödinger bridges

TL;DR

The paper develops a kernel-based framework for Schrödinger bridges by embedding probability measures into an RKHS and penalizing terminal-marginal constraints via the maximum mean discrepancy. It shows that the Schrödinger problem is equivalent to a McKean–Vlasov stochastic control problem with a kernel-based constraint, and proves that epsilon-optimal controls converge to the unique Schrödinger solution. A neural SDE–based algorithm is introduced to solve the resulting MKV control problem, and the approach is validated on 1D and 2D bridging/interpolation tasks. By unifying entropic regularization, kernel embeddings, and neural stochastic dynamics, the work offers a scalable method for bridging arbitrary initial and terminal marginals in diffusion processes.

Abstract

We characterize the Schrödinger bridge problems by a family of Mckean-Vlasov stochastic control problems with no terminal time distribution constraint. In doing so, we use the theory of Hilbert space embeddings of probability measures and then describe the constraint as penalty terms defined by the maximum mean discrepancy in the control problems. A sequence of the probability laws of the state processes resulting from -optimal controls converges to a unique solution of the Schrödinger's problem under mild conditions on given initial and terminal time distributions and an underlying diffusion process. We propose a neural SDE based deep learning algorithm for the Mckean-Vlasov stochastic control problems. Several numerical experiments validate our methods.
Paper Structure (9 sections, 4 theorems, 99 equations, 6 figures, 2 algorithms)

This paper contains 9 sections, 4 theorems, 99 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Hilbert space embeddings of probability measures
  3. Kernel-based approximate bridges
  4. Reduction to Mckean-Vlasov stochastic control problems
  5. Neural SDEs-based methods
  6. Numerical experiments
  7. 1D Bridge: from a Dirac measure to a bimodal distribution
  8. 2D Interpolation of two datasets
  9. Proof of Theorem \ref{['thm:3.1']}

Key Result

Proposition 2.1

Suppose that $(A1)$ holds. Then $\gamma=\gamma_K$ is a metric on $\mathcal{P}(\mathbb{R}^d)$ that metrizes the weak topology.

Figures (6)

  • Figure 4.1: Histograms of optimized $X_t^{(\theta)}$'s for $t=0, 0.33, 0.67, 1$ from left to right. Generated with $2\times 10^5$ samples. The true density $\rho_1$ is plotted in orange.
  • Figure 4.2: The learning curves for $1/\lambda=0.5, 0.05, 0.005, 0.0005$.
  • Figure 4.3: Interpolation of a dataset distribution of two circles and one of a double crescent. The scatter plot of optimized $X_t^{(\theta)}$'s for $t=0, 0.33, 0.67, 1$ from left to right. The target distribution is plotted in orange.
  • Figure 4.4: The learning curves for $1/\lambda=5\times 10^{-2}, 5\times 10^{-4}, 5\times 10^{-6}, 5\times 10^{-8}$, in the case where $\mu_0$ and $\mu_1$ are given by the dataset distributions of two circles and one of a double crescent, respectively.
  • Figure 4.5: Interpolation of two dataset distributions of two circles and one of a double crescent, with different dispersion level. The case of $1/\lambda=5\times 10^{-3}$ and $5000$ epochs. The scatter plot of optimized $X_t^{(\theta)}$'s for $t=0, 0.33, 0.67, 1$ from left to right. The target distribution is plotted in orange.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Definition 5.1
  • Lemma 5.2
  • proof
  • Theorem 5.3
  • proof
  • proof : Proof of Theorem $\ref{['thm:3.1']}$