Schrödinger Bridge 1 D1 (2023.11.02)

n n - Schrödinger Bridge n Schrödinger Bridge n Schrödinger Bridge - 2

3 n (DSM) [Ho, 20][Song 20][Song 21] . . ODE (CNF) , & .

4 n Probability flow [Song et al. 21]: 𝑝! ODE . Fokker-Planck (FP)

5 n [Lipman 22]: CNF ( ) 𝐟𝐭 ( ) , ODE

6

7 ( / ) Waddington landscape [Waddington 1942] : [Tong 23] ( ) : [Liu 22] [Nodozi 23] : https://pubs.acs.org/doi/10.1021/acsnano.9b07849 . ( )

n 8 1. . 2. . 3. . Schrödinger Bridge (SB) [Léonard 13; Chen et al. 21] ( )

9 Schrödinger Bridge

10 SB n Girsanov , Fokker-Planck (FP) ( ) SB

11 SB n SB HJB (Cole-Hopf ) FP - , Schrödinger system (𝜑, & 𝜑) ( ) SB [Pavon & Wakolbinger 91]

12 SB / PDE FP-HJB (𝜓! , 𝑝! ) / PDE - (𝜑! , ' 𝜑! )

13 SB n (Optimal Transport: OT) SB = [Brenier & Benamou 2000] SB SB 𝜖 → 0 OT , .

14 SB n SB ( / ) SB (SOT) [Mikami 08] . HJB

15 Schrödinger Bridge

16 SB IPML [Vargas 21] DSB [De Bortoli 21] DSBM [Shi 23] IDBM [Peluchetti 23] DMSB [Chen 23] Multi-stage SB [Wang 21] NLSB [Koshizuka 22] eAM [Neklyudov 23] SB-FBSDE [Chen 21] DeepGSB [Liu 22] Iterative Proportial fitting (IPF) ( ) SBAlign [Somnath 23] SB-CFM [Tong 23] [SF]^2M [Tong 23] (𝜓! , 𝑝! ) (𝜑! , + 𝜑! ) PIS [Zhang 22] ENOT [Gushchin 22]

17 n , . SB • ( ) • SB / VAE VAE 𝜙 𝜃 𝒩 𝜙 𝜃 𝒩 SB 𝜃 𝑝" 𝑝" 𝑇

18 SB IPML [Vargas 21] DSB [De Bortoli 21] DSBM [Shi 23] IDBM [Peluchetti 23] DMSB [Chen 23] Multi-stage SB [Wang 21] NLSB [Koshizuka 22] eAM [Neklyudov 23] SB-FBSDE [Chen 21] DeepGSB [Liu 22] Iterative Proportial fitting (IPF) ( ) SBAlign [Somnath 23] SB-CFM [Tong 23] [SF]^2M [Tong 23] (𝜓! , 𝑝! ) (𝜑! , + 𝜑! ) PIS [Zhang 22] ENOT [Gushchin 22]

19 SB n Iterative Proportional Fitting (IPF) [Cramar 2000] ( ) SB Half-bridge ( ) IPF , SB / Half-bridge 𝝅𝒕 (𝒊) . [Vargas 21], [De Bortoli 21] mean-matching regression + 𝑢! ($%&') 𝑢! ($%&$) SB Sinkhorn : [Chen, Deng, Fang & Li 23]

20 SB IPML [Vargas 21] DSB [De Bortoli 21] DSBM [Shi 23] IDBM [Peluchetti 23] DMSB [Chen 23] Multi-stage SB [Wang 21] NLSB [Koshizuka 22] eAM [Neklyudov 23] SB-FBSDE [Chen 21 ] DeepGSB [Liu 22] Iterative Proportial fitting (IPF) ( ) SBAlign [Somnath 23] SB-CFM [Tong 23] [SF]^2M [Tong 23] (𝜓! , 𝑝! ) (𝜑! , + 𝜑! ) PIS [Zhang 22] ENOT [Gushchin 22]

21 n Neural Lagrangian Schrödinger Bridge (NLSB) [Koshizuka & Sato 22] , ( . ) ( ) 𝜓! ' 𝑡 = 0 ( 𝑁) & HJB-PDE 𝑡 = 𝑇 ( 𝑀) 𝜆(, 𝜆) : & [Neklyudow 23] (?) SB SB

22 SB IPML [Vargas 21] DSB [De Bortoli 21] DSBM [Shi 23] IDBM [Peluchetti 23] DMSB [Chen 23] Multi-stage SB [Wang 21] NLSB [Koshizuka 22] eAM [Neklyudov 23] SB-FBSDE [Chen 22] DeepGSB [Liu 22] Iterative Proportial fitting (IPF) ( ) SBAlign [Somnath 23] SB-CFM [Tong 23] [SF]^2M [Tong 23] (𝜓! , 𝑝! ) (𝜑! , + 𝜑! ) PIS [Zhang 22] ENOT [Gushchin 22]

23 n SB-FBSDE [Chen, Liu & Theodorou 21] Feynman-Kac (FK) [Exarchos & Theodorou 18] : - Feynman-Kac , SDE . SB Forward-Backward SDEs (FB-SDEs) : PDE

24 n SB-FBSDE [Chen, Liu & Theodorou 21] SB - 𝑣 𝑢! ∗ & 𝑢! ∗ / (𝜑! , # 𝜑! ) , FK , ℎ ℎ PDE FB-SDEs

25 SB n SB-FBSDE [Chen, Liu & Theodorou 21] SB & / [Liu 22] (c.f. NLSB) (𝑍! ', , 𝑍! +) - SDE CNF 1

26 SB IPML [Vargas 21] DSB [De Bortoli 21] DSBM [Shi 23] IDBM [Peluchetti 23] DMSB [Chen 23] Multi-stage SB [Wang 21] NLSB [Koshizuka 22] eAM [Neklyudov 23] SB-FBSDE [Chen 21] DeepGSB [Liu 22] Iterative Proportial fitting (IPF) ( ) SBAlign [Somnath 23] SB-CFM [Tong 23] [SF]^2M [Tong 23] (𝜓! , 𝑝! ) (𝜑! , + 𝜑! ) PIS [Zhang 22] ENOT [Gushchin 22]

27 n Schrödinger Bridge - Conditional Flow Matching (SB-CFM) [Tong et al. 23] SB [Lipman 22] . SB ODE d𝑥 = f, 𝑥 d𝑡 . ( ) SB ( ) 𝜋(𝑥-, 𝑥.) = SB ( 𝜆 = 4𝜖) . 𝜋 , ℒ(𝜃) . SB . ODE / )

28 n SB IPF IPML [Vargas 21] DSB [De Bortoli 21] DSBM [Shi 23] IDBM [Peluchetti 23] DMSB [Chen 23] Multi-stage SB [Wang 21] NLSB [Koshizuka 22] eAM [Neklyudov 23] SB-FBSDE [Chen 21] DeepGSB [Liu 22] ( ) SBAlign [Somnath 23] SB-CFM [Tong 23] [SF]^2M [Tong 23] (𝜓! , 𝑝! ) (𝜑! , + 𝜑! ) PIS [Zhang 22] ENOT [Gushchin 22] / × ×

29 • SB • SB , SB 🔥 • PDE NN • ( , , ) /

30 References [Ho ’21] Ho, J., Jain, A., & Abbeel, P. (2020). Denoising diffusion probabilistic models.Advances in neural information processing systems, 33, 6840-6851 [Song ’20] Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., & Poole, B. (2020, October). Score-Based Generative Modeling through Stochastic Differential Equations. In International Conference on Learning Representations. [Song ’21] Song, Y., Durkan, C., Murray, I., & Ermon, S. (2021). Maximum likelihood training of score-based diffusion models.Advances in Neural Information Processing Systems, 34, 1415-1428. [Lipman ’22] Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., & Le, M. (2022, September). Flow Matching for Generative Modeling. In The Eleventh International Conference on Learning Representations. [Nodozi ’23]Nodozi, I., Yan, C., Khare, M., Halder, A., & Mesbah, A. (2023). Neural Schrödinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly. arXiv preprint arXiv:2307.14442. [Léonard ’13] Léonard, C. (2014). A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems-Series A, 34(4), 1533-1574. [Chen ’21] Chen, Y., Georgiou, T. T., & Pavon, M. (2021). Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schrodinger bridge. Siam Review, 63(2), 249-313. [Mikami ’08] Mikami, T. (2008). Optimal transportation problem as stochastic mechanics. Selected Papers on Probability and Statistics, Amer. Math. Soc. Transl. Ser, 2(227), 75-94.

31 References [Pavon & Wakolbinger ’91] Pavon, M., & Wakolbinger, A. (1991). On free energy, stochastic control, and Schrödinger processes. In Modeling, Estimation and Control of Systems with Uncertainty: Proceedings of a Conference held in Sopron, Hungary, September 1990 (pp. 334-348). Boston, MA: Birkhäuser Boston. [Brenier & Benamou 2000] Benamou, J. D., & Brenier, Y. (2000). A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84(3), 375-393. [Vargas ’21] Vargas, F., Thodoroff, P., Lamacraft, A., & Lawrence, N. (2021). Solving schrödinger bridges via maximum likelihood. Entropy, 23(9), 1134. [De Bortoli ’21] De Bortoli, V., Thornton, J., Heng, J., & Doucet, A. (2021). Diffusion Schrödinger bridge with applications to score-based generative modeling.Advances in Neural Information Processing Systems, 34, 17695-17709. [Shi ’23] Shi, Y., De Bortoli, V., Campbell, A., & Doucet, A. (2023). Diffusion Schr¥" odinger Bridge Matching. arXiv preprint arXiv:2303.16852. [Peluchetti ’23] Peluchetti, S. (2023). Diffusion Bridge Mixture Transports, Schr¥" odinger Bridge Problems and Generative Modeling. arXiv preprint arXiv:2304.00917. [Chen ’23] Chen, T., Liu, G. H., Tao, M., & Theodorou, E. A. (2023). Deep Momentum Multi-Marginal Schr¥" odinger Bridge. arXiv preprint arXiv:2303.01751. [Koshizuka ’22] Koshizuka, T., & Sato, I. (2022, September). Neural Lagrangian Schr¥"{o} dinger Bridge: Diffusion Modeling for Population Dynamics. In The Eleventh International Conference on Learning Representations.

32 References [Neklyudov ’23] Neklyudov, K., Brekelmans, R., Severo, D., & Makhzani, A. (2023). Action Matching: Learning Stochastic Dynamics from Samples. [Chen ’21] Chen, T., Liu, G. H., & Theodorou, E. (2021, October). Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory. In International Conference on Learning Representations. [Liu ’22] Liu, G. H., Chen, T., So, O., & Theodorou, E. (2022). Deep generalized Schrödinger bridge.Advances in Neural Information Processing Systems, 35, 9374-9388. [Wang ’21] Wang, G., Jiao, Y., Xu, Q., Wang, Y., & Yang, C. (2021, July). Deep generative learning via schrödinger bridge. In International Conference on Machine Learning (pp. 10794-10804). PMLR. [Zhang ’22] Zhang, Q., & Chen, Y. (2022). PATH INTEGRAL SAMPLER: A STOCHASTIC CONTROL APPROACH FOR SAMPLING. Proceedings of Machine Learning Research. [Gushchin. ’22] Gushchin, N., Kolesov, A., Korotin, A., Vetrov, D., & Burnaev, E. (2022). Entropic neural optimal transport via diffusion processes. arXiv preprint arXiv:2211.01156. [Somnath ’23]Somnath, V. R., Pariset, M., Hsieh, Y. P., Martinez, M. R., Krause, A., & Bunne, C. (2023). Aligned Diffusion Schrö dinger Bridges. arXiv preprint arXiv:2302.11419. [Tong ’23] Tong, A., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Fatras, K., ... & Bengio, Y. (2023, July). Improving and generalizing flow-based generative models with minibatch optimal transport. In ICML Workshop on New Frontiers in Learning, Control, and Dynamical Systems. [Tong ’23] Tong, A., Malkin, N., Fatras, K., Atanackovic, L., Zhang, Y., Huguet, G., ... & Bengio, Y. (2023). Simulation-free Schrödinger bridges via score and flow matching. arXiv preprint arXiv:2307.03672.