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Schrödinger Bridge問題に基づく拡散生成モデル学習

越塚 毅
November 05, 2023

Schrödinger Bridge問題に基づく拡散生成モデル学習

数値解析と機械学習の協同が拓く新時代の数理科学 (2023.11.02) 講演スライド

越塚 毅

November 05, 2023
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  1. Schrödinger Bridge
    1
    D1
    (2023.11.02)

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

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

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

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  5. 5
    n [Lipman 22]: CNF
    ( ) 𝐟𝐭
    ( )
    ,
    ODE

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  6. 6

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

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

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  9. 9
    Schrödinger Bridge

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

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

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

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

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

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  15. 15
    Schrödinger Bridge

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  16. 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]

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  17. 17
    n
    , .
    SB
    • ( )

    SB
    /
    VAE
    VAE
    𝜙 𝜃
    𝒩 𝜙
    𝜃
    𝒩
    SB
    𝜃
    𝑝" 𝑝"
    𝑇

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  18. 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]

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  19. 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]

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  20. 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]

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  21. 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

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  22. 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]

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

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  24. 24
    n SB-FBSDE [Chen, Liu & Theodorou 21]
    SB
    -
    𝑣
    𝑢!

    &
    𝑢!

    /
    (𝜑!
    , #
    𝜑!
    ) ,
    FK
    ,


    PDE FB-SDEs

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

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  26. 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]

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  27. 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 / )

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  28. 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]
    / ×
    ×

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  29. 29
    • SB
    • SB ,
    SB
    🔥
    • PDE NN

    ( , , ) /

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  30. 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.

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  31. 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
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    References
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    [Liu ’22] Liu, G. H., Chen, T., So, O., & Theodorou, E. (2022). Deep generalized Schrödinger bridge.Advances in Neural Information Processing
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