Schrödinger Bridge Problem

A framework that unifies optimal transport, FB-SDEs, fluid dynamics, and stochastic control.

The classical Schrödinger Bridge Problem (SBP) have found interesting applications in deep generative models (Neal, 2011), (Chen et al., 2022), (De Bortoli et al., 2021) and financial mathematics (Nutz, 2022) (Nutz, 2022) (Henry-Labordere, 2019). The iterative nature in solving this problem shows a great potential to further accelerate the training for score-based generative models, although the later is already the state-of-the-art methods [Image Generation].

The most striking feature of this algorithm is its deep mathematical connections to stochastic optimal control, optimal transport, fluid dynamics (Chen, 2021), and DNN approximations. As such, we have sufficient tools to understand the underlying theories. We refer interested readers to the Appendix A of (Chen et al., 2023).

  1. Neal, R. M. (2011). MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo.
  2. Chen, T., Liu, G.-H., & Theodorou, E. A. (2022). Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory. ICLR.
  3. De Bortoli, V., Thornton, J., Heng, J., & Doucet, A. (2021). Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling. NeurIPS.
  4. Nutz, Z., Wiesel. (2022). Limits of Semistatic Trading Strategies.
  5. Nutz, Z., Wiesel. (2022). Martingale Schrödinger Bridges and Optimal Semistatic Portfolios.
  6. Henry-Labordere, P. (2019). From (martingale) Schrödinger bridges to a new class of stochastic volatility model.
  7. Chen, P., Georgiou. (2021). Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge. SIAM Review.
  8. Chen, Y., Deng, W., Fang, S., Li, F., Yang, N., Zhang, Y., Rasul, K., Zhe, S., Schneider, A., & Nevmyvaka, Y. (2023). Provably Convergent Schrödinger Bridge with Applications to Probabilistic Time Series Imputation. ICML.