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Computational OT #4 - Gradient flow and diffusi...

Computational OT #4 - Gradient flow and diffusion models

A course on computational Optimal Transport

Gabriel Peyré

April 10, 2025
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  1. Gabriel Peyré É C O L E N O R

    M A L E S U P É R I E U R E ∇ Computational Optimal Transport #4 Gradient flow and diffusion models
  2. Generative AI: auto-regressive vs. transport Tell the story, in a

    funny way, of a CNRS researcher presenting generative AI to a large audience. Once upon a time at the CNRS, there was a researcher—let’s call him Dr. Jean- Michel Algorithm—who was bravely preparing to explain generative AI to a packed auditorium of curious minds, skeptics, and that one guy who only came for the free coffee. Jean-Michel strutted onto the stage with the swagger of someone who had just discovered LLMs. Depict a mathematics researcher presenting generative AI in front of a large audience. DALL·E 2
  3. Generative AI: auto-regressive vs. transport Tell the story, in a

    funny way, of a CNRS researcher presenting generative AI to a large audience. Once upon a time at the CNRS, there was a researcher—let’s call him Dr. Jean- Michel Algorithm—who was bravely preparing to explain generative AI to a packed auditorium of curious minds, skeptics, and that one guy who only came for the free coffee. Jean-Michel strutted onto the stage with the swagger of someone who had just discovered LLMs. Depict a mathematics researcher presenting generative AI in front of a large audience. DALL·E 2 Once upon a time at the CNRS there was a researcher, let’s call him Dr. Jean-Michel Algorithm Pre-training: next token prediction. Generation: auto-regressive. Pre-training: denoising. Generation: dynamic transport.
  4. Eulerian vs. Lagrangian representations Evolution of distributions t → αt

    Sampling (flow matching) Optimization (perceptron) Transformers (depth) samples neurons tokens α0 α1 αt x(0) x(t) Genomics (developpement) cells
  5. Eulerian vs. Lagrangian representations Evolution of distributions t → αt

    Sampling (flow matching) Optimization (perceptron) Transformers (depth) samples neurons tokens α0 α1 αt v0 v1 vt x(0) x(t) Eulerian Lagrangian αt vt div(αt vt ) + ∂αt ∂t = 0 Genomics (developpement) cells
  6. Eulerian vs. Lagrangian representations Evolution of distributions t → αt

    Sampling (flow matching) Optimization (perceptron) Transformers (depth) samples neurons tokens Transport map: T : x(0) ↦ x(1), dx(t) dt = vt (x(t)) if then X0 ∼ α0 T(X0 ) ∼ α1 Sampling by transport: α0 α1 αt v0 v1 vt x(0) x(t) Eulerian Lagrangian αt vt div(αt vt ) + ∂αt ∂t = 0 Genomics (developpement) cells
  7. Gradient Structure Riemannian structure at : α ∥v∥2 L2(α) :=

    ∫ ℝd ∥v(x)∥2dα(x) Otto’s Calculus Density manifold Felix Otto John Lafferty Eulerian αt Lagrangian vt ?
  8. Least squares inversion: min vt {∫ 1 0 ∫ ℝd

    ∥vt (x)∥2 dαt (x) dt : div(αt vt ) + ∂αt ∂t = 0} Optimality conditions: vt = ∇ψt ψt = − Δ−1 αt [∂t αt ] Δα ψ := div(α∇ψ) Gradient Structure Riemannian structure at : α ∥v∥2 L2(α) := ∫ ℝd ∥v(x)∥2dα(x) Otto’s Calculus Density manifold Felix Otto John Lafferty Eulerian αt Lagrangian vt ?
  9. Least squares inversion: min vt {∫ 1 0 ∫ ℝd

    ∥vt (x)∥2 dαt (x) dt : div(αt vt ) + ∂αt ∂t = 0} Optimality conditions: vt = ∇ψt ψt = − Δ−1 αt [∂t αt ] Δα ψ := div(α∇ψ) Gradient Structure Riemannian structure at : α ∥v∥2 L2(α) := ∫ ℝd ∥v(x)∥2dα(x) Otto’s Calculus Density manifold Felix Otto John Lafferty Intractable in high dimension. → Leverage specific structure of → αt Flow matching (stochastic interpolant) Optimal transport (displacement interp.) Eulerian αt Lagrangian vt ?
  10. α0 αt α1 Flow Matching and Diffusion Stochastic interpolant (convolution):

    X0 ∼ α0 , X1 ∼ α1 X0 , X1 indepentant . where αt := Law((1 − t)X0 +tX1 )
  11. α0 αt α1 Flow Matching and Diffusion Stochastic interpolant (convolution):

    X0 ∼ α0 , X1 ∼ α1 X0 , X1 indepentant . where αt := Law((1 − t)X0 +tX1 )
  12. α0 αt α1 Flow Matching and Diffusion Stochastic interpolant (convolution):

    X0 ∼ α0 , X1 ∼ α1 X0 , X1 indepentant . where αt := Law((1 − t)X0 +tX1 ) Flow matching: → infvt ∫ ∥x1 − x0 − vt ((1 − t)x0 +tx1 )∥2 dα0 (x0 )dα1 (x1 )dt Prop: vt (x) = 𝔼 X1 X0 [X1 − X0 | x = (1 − t)X0 +tX1] div(αt vt ) + ∂αt ∂t = 0 satisfies x
  13. α0 αt α1 Flow Matching and Diffusion Stochastic interpolant (convolution):

    X0 ∼ α0 , X1 ∼ α1 X0 , X1 indepentant . where min vt {∫ 1 0 ∫ ℝd ∥vt (x)∥2 dαt (x) dt : div(αt vt ) + ∂αt ∂t = 0} Proposition: if , solves α0 = 𝒩 (0,Id) vt (Equivalent to diffusion models) αt := Law((1 − t)X0 +tX1 ) Flow matching: → infvt ∫ ∥x1 − x0 − vt ((1 − t)x0 +tx1 )∥2 dα0 (x0 )dα1 (x1 )dt Prop: vt (x) = 𝔼 X1 X0 [X1 − X0 | x = (1 − t)X0 +tX1] div(αt vt ) + ∂αt ∂t = 0 satisfies x
  14. W2 (α0 , α1 )2 := min T n ∑

    i=1 ∥xi − yT(i) ∥2 Optimal Transport (Wasserstein) Distance ∥xi − yj ∥2 xi yj T Monge 1784
  15. = inf T♯ α0 =α1 ∫ ∥x − T(x)∥2dα0 (x)

    α0 α1 T W2 (α0 , α1 )2 := min T n ∑ i=1 ∥xi − yT(i) ∥2 Optimal Transport (Wasserstein) Distance ∥xi − yj ∥2 xi yj T Monge 1784 General measures: Kantorovitch relaxation Approximation by discrete measures or Kantorovitch 1942
  16. min vt {∫ ∥vt ∥2 L2(αt ) dt : div(αt

    vt ) + ∂t αt = 0} Diffusion model: α0 αt α1 Diffusion vs Optimal Transport αt := Law((1 − t)X0 +tX1 )
  17. min vt {∫ ∥vt ∥2 L2(αt ) dt : div(αt

    vt ) + ∂t αt = 0} Diffusion model: α0 αt α1 Diffusion vs Optimal Transport αt := Law((1 − t)X0 +tX1 ) Dynamic Optimal transport: min vt ,αt {∫ ∥vt ∥2 L2(αt ) dt : div(αt vt ) + ∂t αt = 0} T
  18. min vt {∫ ∥vt ∥2 L2(αt ) dt : div(αt

    vt ) + ∂t αt = 0} Diffusion model: α0 αt α1 Theorem: [Benamou-Brenier] W2 (α0 , α1 )2 = inf T {∫ ∥x − T(x)∥2dα0 (x) : T♯ α0 = α1} αt = ((1 − t)Id + tT)♯ α0 Yann Brenier Jean-David Benamou Gaspard Monge Diffusion vs Optimal Transport =: W2 (α0 , α1 )2 (Wasserstein distance) αt := Law((1 − t)X0 +tX1 ) Dynamic Optimal transport: min vt ,αt {∫ ∥vt ∥2 L2(αt ) dt : div(αt vt ) + ∂t αt = 0} T
  19. αt+τ := arg min α 1 2τ W2 (αt ,

    α)2 + f(α) Implicit Euler step: Wasserstein Gradient Flows Felix Otto David Kinderlehrer Richard Jordan αt αt+τ αt+2τ Optimization over distributions: min α f(α)
  20. αt+τ := arg min α 1 2τ W2 (αt ,

    α)2 + f(α) Implicit Euler step: Single Dirac: αt = δx(t) , x(t + τ) := arg min x 1 2τ ∥x−x(t)∥2 + h(x) h(x) := f(δx ) dx(t) dt = − ∇h(x(t)) Wasserstein Gradient Flows Felix Otto David Kinderlehrer Richard Jordan αt αt+τ αt+2τ x(t) x(t + τ) x(t + 2τ) Optimization over distributions: min α f(α)
  21. αt+τ := arg min α 1 2τ W2 (αt ,

    α)2 + f(α) Implicit Euler step: Single Dirac: αt = δx(t) , x(t + τ) := arg min x 1 2τ ∥x−x(t)∥2 + h(x) h(x) := f(δx ) dx(t) dt = − ∇h(x(t)) Wasserstein Gradient Flows ∂αt ∂t + div(vt αt ) = 0 τ → 0 Proposition: on can take vt = − ∇W f(α) := − ∇ℝd [δf(α)] Frechet derivative: f(α + τβ) = f(α) + τ∫ [δf(α)]dβ + o(τ) Felix Otto David Kinderlehrer Richard Jordan αt αt+τ αt+2τ x(t) x(t + τ) x(t + 2τ) Optimization over distributions: min α f(α)
  22. αt+τ := arg min α 1 2τ W2 (αt ,

    α)2 + f(α) Implicit Euler step: Single Dirac: αt = δx(t) , x(t + τ) := arg min x 1 2τ ∥x−x(t)∥2 + h(x) h(x) := f(δx ) dx(t) dt = − ∇h(x(t)) Wasserstein Gradient Flows ∂αt ∂t + div(vt αt ) = 0 τ → 0 Proposition: on can take vt = − ∇W f(α) := − ∇ℝd [δf(α)] Frechet derivative: f(α + τβ) = f(α) + τ∫ [δf(α)]dβ + o(τ) Felix Otto David Kinderlehrer Richard Jordan Linear: f(α) = ∫ h(x)dα(x) ∇W f(α) = ∇h Neg-entropy: f(α) = ∫ log(dα dx (x))dα(x) ∇W f(α) = ∇log(dα dx ) Quadratic: f(α) := ∬ k(x, y)dα(x)dα(y) ∇W f(α) = 2∫ ∇x k(x, y)dα(y) ∂αt ∂t = Δαt dx(t) dt = − ∇h(x(t)) Interacting particles αt αt+τ αt+2τ x(t) x(t + τ) x(t + 2τ) Optimization over distributions: min α f(α)
  23. Linear vs Non-linear Diffusions <latexit sha1_base64="SPs/Y8MbayYWNRJ0nuvd33h2Vik=">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</latexit> Linear/advection: αt · x(t)

    = − ∇h(x) f(α) = ∫ hdα ∂t α = div(α∇h)
  24. Linear vs Non-linear Diffusions <latexit sha1_base64="SPs/Y8MbayYWNRJ0nuvd33h2Vik=">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</latexit> Linear/advection: αt · x(t)

    = − ∇h(x) f(α) = ∫ hdα ∂t α = div(α∇h)
  25. Linear vs Non-linear Diffusions <latexit sha1_base64="Zr2FwZcY0s8qtfjTvFGZ16X0YpU=">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</latexit> @t↵ = ↵ <latexit

    sha1_base64="k4D99DrG+IKF5b//kMXXkqu7NEc=">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</latexit> f(↵) = R log(d↵ dx )d↵ <latexit sha1_base64="CgiIBNu5Aft/Z53Awd9qhnznPms=">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</latexit> f(↵) = 1 p 1 R (d↵ dx )pd↵ <latexit 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@t↵ = ↵p <latexit 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Linear/advection: αt · x(t) = − ∇h(x) f(α) = ∫ hdα ∂t α = div(α∇h)
  26. Crowd motion with congestion https://www.youtube.com/watch?v=tDQw21ntR64 Tim Whittaker (New Zealand) <latexit

    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C , ↵ ; d↵ dx < T <latexit 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Congestion constraint: <latexit 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r' <latexit 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[Maury, Roudne↵-Chupin, Santambrogio] <latexit sha1_base64="OCaEP+M/QxoslcIqs6+U1e2zcCw=">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</latexit> f(↵) = R 'd↵ + R log(d↵ dx )d↵ + ◆C(↵)
  27. Crowd motion with congestion https://www.youtube.com/watch?v=tDQw21ntR64 Tim Whittaker (New Zealand) <latexit

    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C , ↵ ; d↵ dx < T <latexit 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Congestion constraint: <latexit 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r' <latexit 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[Maury, Roudne↵-Chupin, Santambrogio] <latexit sha1_base64="OCaEP+M/QxoslcIqs6+U1e2zcCw=">AABFOnictVzrcxS5EReX14W8uORjvszFkIIcIcYhj6qrqzqwjfHhA8OuDXcsuGZ3Z9cL451lH+axt/9ZKvk78jWVVOUqleRL/oD0QxppdjXTGocwZVuj0a+71SO1ulsa2qN0MJmur//53Hvf+Oa3vv2d9797/nvf/8EPf3Thgx8fTrLZuJMcdLI0Gz9ux5MkHQyTg+lgmiaPR+MkPmmnyaP2i018/ug0GU8G2bA5fTNKnp7E/eGgN+jEU6g6uvD4Yu9yK043r0SfRK3BcBq1RseDqNXFuugjXZVm/cut3jjuzPnBAv5GrxdXoF2E99gum8ZH89ZmZ8HkLh5dWFu/tk7/otXCdV1YU/rffvbBh39TLdVVmeqomTpRiRqqKZRTFasJXE/UdbWuRlD3VM2hbgylAT1P1EKdB+wMWiXQIobaF/C7D3dPdO0Q7pHmhNAd4JLCzxiQkboEmAzajaGM3CJ6PiPKWFtGe040UbY38LetaZ1A7VQdQ62EMy1DcdiXqeqp31MfBtCnEdVg7zqayoy0gpJHTq+mQGEEdVjuwvMxlDuENHqOCDOhvqNuY3r+d2qJtXjf0W1n6muS8hJckWro3mc5hVidEv2I3uYMnrE8KXDuA4VE9xFLr0jXJ9T7IbSfQ/09uBZUMjppwzWn2kUlchMuH3JTRO7A5UPuiMg9uHzIPRG5D5cPua+RiB2Tzv34Blw+fEPk/AAuH/KBiHwIlw/5UEQewuVDHorIL+HyIb8Ukbfh8iFvi8i7cPmQd0VkEy4fsikiD+DyIQ9E5DZcPuS2RpbP1DFcGdEZCLPyJpSLPNBSpFBzU5TvFllHH/ZWwJzulGDlWb0Ff/3YrQCdJiXY7YBx1yvByiNvB2ykHyvboju0mviwd0TsLowAP3ZXxH6mnpdgPwuYaS9KsPJc24N2fqxsfT+HOz/2cxF7D0p+rLxG3YcaP/Z+wIoxKsHui9gH6mUJNsTqj0uwst1vgF3xY+V1qgnt/dgQazorwcr29BA8GD9WXq0eQa0f+0jEPlavS7CPRewXYN392C8CVti3JVizxp6nFaRP/kgCM7aKWpzPSiyNgFos8E/ztSUl37gN9RKmn2P6hDkRETs5YicQsZcj9oLlmuR2dEL+rsylkSMagYh2vjZhaSq27+btsZQGILZyxNYSosojxXdt+nJK3oWpkZDTfOXCUkifstx+YynR46Ha8hrE/QKCx/YxjfyrFC1hBIWaqqJ2nK/xjIzovgrxiqI300vDQ8ZNc6vgol6LqLYH1RZRbzyoNyJq5kHNRNSpB3UqouzMd3GtgBFg9Y/vYk53PALYRy6/IvAKbsKqcwfmaATjZx+8wIdUcx/+Nij2lq4qyTCax3USsxxPC5Z4DKW5WoN6GxVuUXyd0gxLQDJueV/H+HiHuY25nnNshRf5Sh7lGZNwOgOSp5/TQW8xovlUj85dqlmQd8elevg7+bw3pXr4bdL4grx4LtXDT7X00zPI3tTY5hmwDZhNI619W65Lg/MvTMOUz9OqixYX3+qJHjNI73VN+rv6zeye4b1sUon1Y8v1aEyc/k0K/atDw+p54ui5HhX0ntjrNaWodk+GOu615boyZLSKDrUc9q7um8E2Xf1mTLkejX3wuDYp5p475bqjd5T3xpbr0ThUnPdckCdvyvVo9Ome9WHL9WhgtiXWcb4t17XsqAGOnW25rlUfUhYYc0A85rnGekVj8pNmmtqA/IPqbI3r86+uY5izeZbHCNWUrG9bTqedr2XVEhl/IQGrNq0pB/oXM8cHK9KYqw0xvmIZpoX1fZWOXeNR83ugxQhmP+8BSDnzFCQ0OQm03ilQvC5GXcWeGdyGiMNR0ltCtXTtVPQWLV/OGhXrjqhWistsb60eW2SvJzT2RuQT7pFmJT3slb7hMoqShvYKGpLp1dHdWz1fi9pfF3GjJcQoH2kd2hHinbTqONWn9Yaj40t6l2cKF+/52PGL2eaetjYY82Rki1CWKp5uO5NHcutwXb2qbI6bn0X0RtFenZLVGNCO1ESMQk22mL3xOd1b2ge0J4c8mEYH3mOkqYwU75phFh3z6RFZVNfeSrxRXyZDx+UJWV1jj6vRfQfd96DrxzibsGLcg1ITYoYDuGsGRDnnc11lpPGx+mW+O5rRG6yO6NOChTQ02N4kBQtZFWUfF6i8AjSOBo7Sw2ks0zH41golOer3yWNj16Llv0Q7t2Z/O6YxXj6ayzMxXeK6QVwjmjW8q8t3yxxYgrn3yQb5r9W9RH51OKINlbg+czizXoa0459QBDsizzil2SbNjmJrNz+1/MRw2ldm7xx3szOykBHZvwjWp4zGZEQ/7tkBs4POFiElGxlidwa5d+PzdQbiGLN+3EDxqQY73hKyZTPib+i6s2tCY5EjBl4HFktj2+hkj3zBhLiOtXW3c7t69UGkPSfhjhKmaMfKZeJ/hX6bHzNO1lZGBGoY38BE2zrf+8goZkEdxbTKV9sg09aV8mIuwzMttV3/rEwXC5JtUcSF8uBq3QXOHbpnXjhKxiT3ZKUNr6NV2VykPFrSI/a2R1E82/2+XoFR7qu0Sq7RnGvRKOnDKJjmUYRpK2WRl/lW8ypSD6M9+b9Qt7ouag0pRspmcFlDUn4/oWjNlTKFUc3j9wXNJr/Wx0utqvkMaSyeOHP5K6j9EH4buc19GJ12wSrcojHAFOyd1QjXRCstwnjdKvAyI9PQsveWnx2TppVbc5b4mq2bjbFPa1PZp1HzWmctTPksNJ47NJ4H6rBJe41Wi6beWKIjMbZo6t3KUH51uDVrUJ6JlGWPzKAGAVK6sVQY1a5IVY7xDeqtSGtdpBXDbHV3A9w5H4L0z/Xl2f1VvrpH6jb5Nh3ywDh+6dIsHZDPZWqrIzWmgJxvaPvqzv4W1SD3NllQpMznOHHG8K5Th65FLunP9cqWkZ23FsGcW3ql2xgb26Lyr1eQJzQnJjQvDeIGtUi0/K4c0ZJFuub4HBFl/mPyqdjvqI6Z3db2nUQFf8LGmzyrLC+OFIakfynztrsSve468WtEMeFMe9dtoFX/DSMFxphMgt+znNAbwlWOdxLYo22T/Vy1U7yLN3QkukZSz9UnATaGo1471t2xZXps+vYLaIlat2/d10LmlwZzlPidZUcvplXtRPuo86X7s9GK9SpXvK/Sw2yJr9XHjNq4kYWN8oqYlvo4mAtLVI8LY0K41OtFHfnrSV5HZt6dCqVsWhvKxUwD25hjipekc6CI8Hl3l73e3BWhH+0Vem3CutS4RqKE2bhM5wdcS4tZqWgpQnLrpTUpddajsvXC8nBXDWvH2VImZAVTJeVuuLXbh1YhWpGzMUyho/hkb1mc6NL8GC78HSlflGg4huQQG+Dn3lSbavsdnIp4qcuc2YyoBm1CdykGj3U/iy2qdfTSoe7SD+EQzmMAupakH9CKWld2pixL7lIPp/+KrMFYJaL0tmX9Prhc5J6scqrTnwFZOLk3A2W+yanbF8MhpCdFLuF8eH9D6kVPmW+b6vXBUJd7UORQh4c5zxD2zm3r+rxcTtX6WuUSyoPXAbPzYnC4A1ges9h2IRZq7LyRd88BrUOvgrpZLf7Xfhg+llN9XqHcJvTN2fOAt87tEp2ZRb+4/pyx3EJGcznHcJ5Z3jvrNfn5sf8X1XpTmdObd08f/VI7BgyvueJ8qCwd491RZOUNpYL7Az4ZMvVv9adz8lcJL3MaZXLUoWT2K8qpmRYyNfPlpa935lmITJZOmUxFajaeaNDJ2E21q27Dz2buAdY9JcrfVPJfxPq/o+1CbY+sh8mmcwahRXUJZUHsblqX7u052jKJ8Uwvn/FtQg3uie9RLZ73vUft8cxvs9C38i9JeK5/rjLVLUQmy7t8dl61oQfFHTjOBZnvfSM6U8/ZLD6BdhKwx8jnqDhSMl8/zwnRpbhwWdI5IcxoqaLc9lJu05mkpIR2u9C3Do3wkd7px30HPJ8f59mlSP2K6mK9OuBKLUm175HqCWUG2qT/dYjQfqOuwt+ruuyXdH9F0gm9g6JEr51n1SfBFt5xYb9mvER5MJOpO9XtMorq7e5hdSZ2q5QLn3ivxvcr8H1Hyga9rRcUd49Vde5wVkFzpmVy93OHyuQ9WQ8Yzcb5+KiOn08reJ0G9P9uKfquI+kOyNKmbHtE+3ljopdq3WyT9Hyusjpve6dCWvPVJtO0JyvtODBnJKv3BFI97spnP5+DlHI1SQkdd67ziUzptMjAS0men6OA0xBxQG/lvob0VKIyEyWZBXyJfBogy2kAnZ4gTU+k0Bcl0fbh6MLa9eX/62O1cLhx7fpvr914sLH26S39/4C8r36qfqYuw9r3O/UpjP99dQCc/qj+qv6p/rX1h62/bH299Q9u+t45jfmJKvzb+s9/AYBoX/U=</latexit> f(↵) = R 'd↵ + R log(d↵ dx )d↵ + ◆C(↵)
  28. Training Dynamics of 2 layers MLP 2 layers perceptron: z

    ↦ ∑n k=1 uk σ(⟨z, uk ⟩) σ (uk )k (vk )k z
  29. min u1 ,v1 ,… F(u1 , v1 , …) Training

    d(ui , vi ) dt = − ∇F(u1 , v1 , …) := ∑ i ( n ∑ k=1 uk σ(⟨zi , vk ⟩) − yi)2 Training Dynamics of 2 layers MLP 2 layers perceptron: z ↦ ∑n k=1 uk σ(⟨z, uk ⟩) σ (uk )k (vk )k z (uk , vk )
  30. min u1 ,v1 ,… F(u1 , v1 , …) Training

    d(ui , vi ) dt = − ∇F(u1 , v1 , …) := ∑ i ( n ∑ k=1 uk σ(⟨zi , vk ⟩) − yi)2 Training Dynamics of 2 layers MLP 2 layers perceptron: z ↦ ∑n k=1 uk σ(⟨z, uk ⟩) σ (uk )k (vk )k z f(α) = ∫ kdα ⊗ α + ∫ hdα := ∑ i (∫ uσ(⟨z, v⟩)dα(u, v) − yi)2 α = ∑ k δ(uk ,vk ) ∂αt ∂t − div(∇W f(αt ) αt ) = 0 min α f(α) α (uk , vk ) k(u, v, u′  , v′  ) := ∑ i uu′  σ(⟨zi , v⟩)σ(⟨zi , v′  ⟩)
  31. min u1 ,v1 ,… F(u1 , v1 , …) Training

    d(ui , vi ) dt = − ∇F(u1 , v1 , …) := ∑ i ( n ∑ k=1 uk σ(⟨zi , vk ⟩) − yi)2 Training Dynamics of 2 layers MLP 2 layers perceptron: z ↦ ∑n k=1 uk σ(⟨z, uk ⟩) Theorem: for perceptrons, if has enough neurons, can only converge to a global minimum. αt=0 αt « Global » convergence, despite not being convex. → F Lenaic Chizat Francis Bach σ (uk )k (vk )k z f(α) = ∫ kdα ⊗ α + ∫ hdα := ∑ i (∫ uσ(⟨z, v⟩)dα(u, v) − yi)2 α = ∑ k δ(uk ,vk ) ∂αt ∂t − div(∇W f(αt ) αt ) = 0 min α f(α) α (uk , vk ) k(u, v, u′  , v′  ) := ∑ i uu′  σ(⟨zi , v⟩)σ(⟨zi , v′  ⟩)
  32. . . . . . . . . . .

    . . . . . . . OT-based joint dimensionality reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OT-based trajectory inference . . . . . . . . . . . . Conclusion Zebrafish embryos along development Sampling Optimizing Genomics Transformers
  33. 1013 cells in the human body, with vastly different functions.

    Early efforts to cartography cell identity relied on microscopy1. Recent initiatives me molecular profile of s in the human body, ly different functions. Early efforts to cartography cell identity relied on microscopy1. Recent initiatives measure th molecular profile of the cell e human body, erent functions. Early efforts to cartography cell identity relied on microscopy1. Recent initiatives measure the molecular profile of the cell2. ~ cells in the human body, with vastly different functions. 1013 Early efforts to cartography cell identity relied on microscopy. [Ramón y Cajal, 1899] Recent initiatives measure the molecular pro fi le of the cell. [Regev et al., eLife, 2017] Unraveling cell diversity
  34. Single Cell Multi-omics Understanding cell diversity: many types, many states

    for each type. Applications: cancer mutations, dynamic of adaptation, development, … <latexit 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cells <latexit 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2 Rd Tissue Dissociation isolation RNA amplification sequencing … <latexit sha1_base64="6L8H4Gt+fXCCPCipj/akDBzsUqw=">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</latexit> d ⇠ 102
  35. Single Cell Multi-omics Understanding cell diversity: many types, many states

    for each type. Applications: cancer mutations, dynamic of adaptation, development, … <latexit 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cells <latexit 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2 Rd Tissue Dissociation isolation RNA amplification sequencing … <latexit 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d ⇠ 102 Multi-omics integration: next frontier … Accessability Gene Expression A B C A B C A (n) A (n) A (n) A (n) A (n) Protein A (n) RNA Chromatin Protein Abundance A D E Gene Expression A B C D E F G RNA DNA Protein Transcription Translation A (n) A (n) A (n) A (n) A (n) Protein A (n) RNA Chromatin Protein Abundance A D E Gene Expression A B C D E F G ATAC-seq Different omics spaces: RNA-seq CITE-seq <latexit 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d ⇠ 104 <latexit 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d ⇠ 105
  36. he joint profiling of gene expression and urface proteins enabled

    to identify a new ubpopulation of CD8 TEM cells8. Spatial transcriptomics profiled across time have allowed to study development at unprecedented resolution9. CD103+ CD8+ TEM CD8+ TEM Zebrafish embryos along development ing of gene expression and ns enabled to identify a new of CD8 TEM cells8. Spatial transcriptomics profiled across time have allowed to study development at unprecedented resolution9. CD103+ CD8+ TEM Zebrafish embryos along development The joint pro fi ling of gene expression and surface proteins enabled to identify a new subpopulation of CD8 TEM cells. [Hao et al., Cell, 2021] Spatial transcriptomics pro fi led across time have allowed to study development at unprecedented resolution. [Liu et al., Developmental Cell, 2022] Examples of recent biological discoveries
  37. Comparing Distributions for Single Cells • Until recently, samples contained

    many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 carelli et al. 2018 2Lähnemann et al. 2020 Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 <latexit sha1_base64="8h9qfn6MXuBvKOcg4p0NkYUZouc=">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</latexit> Bulk <latexit sha1_base64="xfWO8cpwoNz5R1B8gV9utdLBqVI=">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</latexit> Single cell
  38. Comparing Distributions for Single Cells • Until recently, samples contained

    many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 carelli et al. 2018 2Lähnemann et al. 2020 Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 <latexit sha1_base64="8h9qfn6MXuBvKOcg4p0NkYUZouc=">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</latexit> Bulk <latexit sha1_base64="xfWO8cpwoNz5R1B8gV9utdLBqVI=">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</latexit> Single cell <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">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</latexit> ? <latexit sha1_base64="QNHLtJL77Xel7kqDwcr8uqkXoFI=">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</latexit> Cluster cells from a single biopsy. <latexit sha1_base64="NAYrO+hblycchNy+rUEIB63nwtY=">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</latexit> “Distance” between cells? <latexit sha1_base64="yFN6xhMLWSFTo6y4Ic3WSee5g64=">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</latexit> OT on genes’ space. <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">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</latexit> ?
  39. Comparing Distributions for Single Cells • Until recently, samples contained

    many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 carelli et al. 2018 2Lähnemann et al. 2020 Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 <latexit sha1_base64="8h9qfn6MXuBvKOcg4p0NkYUZouc=">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</latexit> Bulk <latexit sha1_base64="xfWO8cpwoNz5R1B8gV9utdLBqVI=">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</latexit> Single cell <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">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</latexit> ? <latexit sha1_base64="QNHLtJL77Xel7kqDwcr8uqkXoFI=">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</latexit> Cluster cells from a single biopsy. <latexit sha1_base64="NAYrO+hblycchNy+rUEIB63nwtY=">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</latexit> “Distance” between cells? <latexit sha1_base64="yFN6xhMLWSFTo6y4Ic3WSee5g64=">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</latexit> OT on genes’ space. <latexit sha1_base64="c8Nzu7i/2HdZbRFNoq24LWiMXCg=">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</latexit> Match cells between two biopsies. <latexit sha1_base64="smAhPpTgHUrzBKRMXffs73pgv0w=">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</latexit> OT on cells’ space. <latexit sha1_base64="pydPpuNuJAJzk3J9ohxA5exGGlo=">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</latexit> ?
  40. Context and aims OT cell-cell similarity Wasserstein Singular Vectors Paired

    multiomics integration Thesis overview Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 OT for Cell to Cell Dissimilarity b <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 a <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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sparse <latexit 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noisy
  41. Context and aims OT cell-cell similarity Wasserstein Singular Vectors Paired

    multiomics integration Thesis overview Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 OT for Cell to Cell Dissimilarity b <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 a <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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noisy <latexit 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min P1=a,P>1=b P i,j d(xi, xj) Pi,j OT(a, b) :=
  42. Context and aims OT cell-cell similarity Wasserstein Singular Vectors Paired

    multiomics integration Thesis overview Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 OT for Cell to Cell Dissimilarity b <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Single-cell profiling uncovers cellular heterogeneity • Until recently, samples contained many cells (bulk omics) • Today, we can measure omics at the single cell level1. 2 1Mincarelli et al. 2018 2Lähnemann et al. 2020 a <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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OT <latexit 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Bio-inspired <latexit 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Data-driven <latexit 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(e.g. via regulation networks) <latexit 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Choice of <latexit 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gene’s distance <latexit 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d(xi, xj) <latexit 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d(xi, xj) <latexit 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sparse <latexit 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noisy <latexit sha1_base64="rckUlp4AuIWgbmmyBNBnPVGh++g=">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</latexit> min P1=a,P>1=b P i,j d(xi, xj) Pi,j OT(a, b) :=
  43. Paired Multi-omics Integration > pip install mowgli <latexit sha1_base64="D+BTpQDTLsuQxDGkm5bQcuzxV/s=">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</latexit> ⇡

    <latexit 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⇥ <latexit sha1_base64="D+BTpQDTLsuQxDGkm5bQcuzxV/s=">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</latexit> ⇡ Geert-Jan Huizing Laura Cantini wk αk ATAC ATAC H αk RNA RNA H min (wk ),(Hm ) {∑ m ∑ k OT(Hm wk , αm k ) : wk ≥ 0,Hm ≥ 0}
  44. Paired Multi-omics Integration > pip install mowgli <latexit sha1_base64="D+BTpQDTLsuQxDGkm5bQcuzxV/s=">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</latexit> ⇡

    <latexit 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⇥ <latexit 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⇡ <latexit sha1_base64="4zRMyofGVvZDzHMLkoMkcIwcgkU=">AABE6nictVzbchu5EYU3t41z8yaPeZmN1ilvylFkxU5StZWqtUVZ1lpryyYle23ZLl5G9NgjDs0h6QtXP5HKSyqVPOU/8h35gFQlT/mF9AUYYEjMNEZxNCURA+J0N3qARncDo944TfLpxsY/zn3wjW9+69vf+fC757/3/R/88EcXPvrxYZ7NJv34oJ+l2eRhr5vHaTKKD6bJNI0fjidx96SXxg96L7fw+wfzeJIn2agzfTuOn5x0h6PkOOl3p1D1qJ/O8mk8SUbDZxfWNtavbcDPZrSxjp9Xr0W65lp0hWo2NtaU/tnPPvr4n+pIDVSm+mqmTlSsRmoK5VR1VQ7XY3VFbagx1D1RC6ibQCmh72N1qs4DdgatYmjRhdqX8HcId4917QjukWZO6D5wSeF3AshIXQRMBu0mUEZuEX0/I8pYW0V7QTRRtrfw2dO0TqB2qp5DrYQzLUNx2JepOla/oz4k0Kcx1WDv+prKjLSCkkdOr6ZAYQx1WB7A9xMo9wlp9BwRJqe+o2679P2/qCXW4n1ft52pf5OUF+GKVFv3PisodNWc6Ef0NGfwHcuTAuchUIh1H7H0mnR9Qr0fQfsF1N+B65RKRic9uBZUe1qL3ILLh9wSkTtw+ZA7InIPLh9yT0Tuw+VD7mskYiekcz++DZcP3xY534PLh7wnIu/D5UPeF5GHcPmQhyLyEVw+5CMReRMuH/KmiLwNlw95W0R24PIhOyLyAC4f8kBEbsPlQ25rZPVMncCVEZ1EmJXXoVzmgZYihZrronw3yDr6sDcC5nS/AivP6hZ8+rGtAJ3GFdjtgHF3XIGVR94O2Eg/VrZFt2g18WFvidhdGAF+7K6I/UK9qMB+ETDTXlZg5bm2B+38WNn6fgl3fuyXIvYOlPxYeY26CzV+7N2AFWNcgd0XsffUqwpsiNWfVGBlu98Gu+LHyutUB9r7sSHWdFaBle3pIXgwfqy8Wj2AWj/2gYh9qN5UYB+K2K/AuvuxXwWssO8qsGaNPU8ryJD8kRhmbB21bjErsTQGal2Bf1qsLSn5xj2olzDDAjMkzImI2CkQO4GIvQKxFyxXXtjRnPxdmUu7QLQDEb1ibcLSVGw/KNpjKQ1AtApEawlR55HiszZ9mZN3YWok5LRYubAU0qessN9YivV4qLe8BnG3hOCx/ZxG/mWKljCCQk3VUXterPGMjOi+DvGaojfTS8NDxk0Lq+Ci3oiongfVE1FvPai3ImrmQc1E1NyDmosoO/Nd3FHACLD6x2exoDseAewjV18ReAXXYdW5BXM0gvGzD17gfaq5C59tir2lq04yjOZxncQsx5OSJZ5AaaHWoN5GhS2Kr1OaYTFIxi3v6hgf7zC3sdBzjq3wabGSR0XGJJxOQvIMCzroLUY0n5rRuU01p+TdcakZ/lYx702pGX6bNH5KXjyXmuGnWvrpGWTvaGznDNg2zKax1r4tN6XB+RemYcrnadVFi4tP9USPGaT3piH9Xf1kds/wXLaoxPqx5WY0cqd/eal/TWhYPeeOnptRQe+JvV5Tihr3ZKTjXltuKkNGq+hIy2Hvmj4ZbDPQT8aUm9HYB49ri2LuhVNuOnrHRW9suRmNQ8V5z1Py5E25GY0h3bM+bLkZDcy2dHWcb8tNLTtqgGNnW25q1UeUBcYcEI95rrFe0YT8pJmmlpB/UJ+tcX3+1XUMczZPixihnpL1bavp9Iq1rF4i4y/EYNWmDeVA/2Lm+GBlGgu1KcZXLMO0tL6v0rFrPGp+D7QYweznPQApZ56ChCYngdY7BYpXxKir3DOD2xRxOEqOl1BHunYqeouWL2eNynXPqFaKy2xvrR6PyF7nNPbG5BPukWYlPexVPuEqipKG9koakuk10d07PV/L2t8QceMlxLgYaX3aEeKdtPo41af1tqPji3qXZwoX7/nY8YvZ5mNtbTDmycgWoSx1PN12Jo/k1uG6elnZHDd/F9ETRXs1J6uR0I5ULkahJlvM3viC7i3tA9qTQx5Mow/PMdJUxop3zTCLjvn0iCyqa28l3qgvk6Hjck5W19jjevTQQQ896OYxzhasGHeg1IGY4QDuOgFRzvlCVxlpfKJ+WeyOZvQE6yP6tGQhDQ22N3HJQtZF2c9LVF4DGkcDR+nhNJbpGPzRCiU56vfJY2PXsuW/SDu3Zn+7S2O8ejRXZ2IGxHWTuEY0a3hXl++WObAEC+83m+S/1vcS+TXhiDZU4vrU4cx6GdGOf0wR7Jg845RmmzQ7yq3d/NTyN4bTvjJ757ibnZGFjMj+RbA+ZTQmI/p1zw6YHXS2CCnZyBC7kxTejc/XScQxZv24RPGpBjveYrJlM+Jv6LqzK6exyBEDrwOnS2Pb6GSPfMGYuE60dbdzu371QaQ9J+GOEqZox8ol4v8p/TW/ZpysrYwI1DA+gVzbOt/zyChmQR11aZWvt0GmrSvlJ4UMT7XUdv2zMn1SkqxFERfKg6v1ADj36Z554SiZkNz5ShteR+uyuUh5vKRH7O0xRfFs94d6BUa5L9MquUZz7ohGyRBGwbSIIkxbKYu8zLeeV5l6GO38/0Ld6rqsNaQYKZvBZQ1J+f2YojVXyhRGNY/flzSb/FqfLLWq5zOisXjizOWvofZj+GvkNvdhdHolq3CDxgBTsHdWI1wTrbQI43WjxMuMTEPL3lt+dkyaVm7NWeJrtm42xp43prJPo+aNzlqY8llovHBovAjUYYf2Gq0WTb2xRM/E2KKjdytD+TXh1mlAeSZSlj0yg0oCpHRjqTCqA5GqHOMb1DuR1oZIqwuz1d0NcOd8CNI/15dn99fF6h6pm+Tb9MkD4/hlQLM0IZ/L1NZHakwBOV/V9tWd/UdUg9x7ZEGRMp/jxBnDu059uk4LSX+uV7aM7Ly1CObc0mvdxtjYIyr/egV5QnMip3lpEFepRazld+WIlizSuuNzRJT575JPxX5HfczstrbPJCr5Ezbe5FlleXGkMCL9S5m33ZXoddeJXyOKCWfau+4BreZPGCkwxmQS/J5lTk8IVzneSWCPtkf2c9VO8S7eyJFonaReqN8H2BiOeu1Yd8eW6bHp2y+gJWrdPnVfC5lfGsxR4neWHb0urWon2kddLN2fjVZXr3Ll+zo9zJb4Wn3MqI0bWdgor4w5Up8Fc2GJmnFhTAiXZr1oIn8zyZvIzLtToZRNa0O5nGlgG/Oc4iXpHCgifN7dJa8396nQj94KvR5hXWpcI1HCbFym8wOupcWsVLQUIbn10pqUOutR1XphebirhrXjbCljsoKpknI33Nrtw1EpWpGzMUyhr/hkb1Wc6NL8DC78GylflGg4huQQ2+DnXldbavs9nIp4pcuc2YyoBm3CYCkG7+p+llvU6+iVQ92lH8IhnEcCupakT2hFbSo7U5Yld6mH039N1mCiYlF627J5H1wuck9WOTXpT0IWTu5Nosw7OU37YjiE9KTMJZwP729IvThW5t2mZn0w1OUelDk04WHOM4Q9c9u6OS+XU72+VrmE8uB1wOy8GBzuAFbHLLZdiIWaOE/k/XNA63BcQ92sFv9rPwwfy6k5r1BuOb1z9iLgqXO7WGdm0S9uPmcst5DRXM0xnGdW9M56TX5+7P9FjZ5U5vTm/dNHv9SOAcNroTgfKkvHeHcUWXlDqeD+gE+GTP1H/f2c/FbCq4JGlRxNKJn9impqpoVMzbx56eud+S5EJkunSqYyNRtPtOlk7JbaVTfhd6vwAJueEuV3KvkTsf73aAdQe0zWw2TTOYNwRHUxZUHsbtqA7u052iqJ8Uwvn/HtQA3uie9RLZ73vUPt8cxvp9S36jdJeK5/qTI1KEUmy7t8dl71oAflHTjOBZn3fSM6U8/ZLD6BdhKwx8jnqDhSMm8/LwgxoLhwWdIFIcxoqaPc81Lu0ZmkuIJ2r9S3Po3wsd7px30HPJ/fLbJLkfoV1XX16oArtSTVvkeqx5QZ6JH+NyBCu6Yuw+dlXfZLur8iaU7PoCzRG+e7+pNgp95xYd9mvEh5MJOpm+t2GUX1dvewPhPbquTCJ97r8cMa/NCRsk1P6yXF3RNVnzuc1dCcaZnc/dyRMnlP1gNGs91ifNTHz/MaXvOA/t+uRN92JN0BWXqUbY9oP29C9FKtm22Sns9V1udtb9VIa97aZJr2ZKUdB+aMZP2eQKrHXfXs53OQUq4mrqDjznU+kSmdFkm8lOT5OQ44DdEN6K3c15CeSlRmoiSzgDeR5wGyzAPoHAvSHIsUhqIk2j48u7Bm/sVHVF043Fy/8pv1q/c21z6/of8PyIfqp+pn6hKsfb9Vn8P431cHwGmk/qj+ov7aSlt/aP2p9Wdu+sE5jfmJKv20/vZfLKxEFw==</latexit> clustering Geert-Jan Huizing Laura Cantini wk αk ATAC ATAC H αk RNA RNA H min (wk ),(Hm ) {∑ m ∑ k OT(Hm wk , αm k ) : wk ≥ 0,Hm ≥ 0}
  45. Paired Multi-omics Integration > pip install mowgli <latexit sha1_base64="D+BTpQDTLsuQxDGkm5bQcuzxV/s=">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</latexit> ⇡

    <latexit 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⇥ <latexit 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⇡ <latexit 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clustering <latexit sha1_base64="lIRSBtvJcg/dPYE+yx9m6aNrdEs=">AABE6XictVzbchTJES3WtzW+sfajX3qtxcE6WHnQwhpiwxELGiG0CBDMSMAiIObSGgb1XJieGS6z+giHXxwO+8kf4u/wBzjCfvIvOC9VXdUz1Z3VMqZDUnV1ncys7KqszKxq2uOkn05rtX+c+eA73/3e93/w4Q/P/ujHP/npz8599PODdDSbdOL9zigZTR61W2mc9Ifx/rQ/TeJH40ncGrST+GH7eBOfP5zHk7Q/Gjanb8fx00GrN+wf9TutKVQ93o6HcZTG0+j5ubXa+rVrtcuXrkW19Su12sbVL6BQ+3zj6pUr0aX1Gv1bU/rf3uijj/+pDlVXjVRHzdRAxWqoplBOVEulcD1Rl1RNjaHuqVpA3QRKfXoeqxN1FrAzaBVDixbUHsPvHtw90bVDuEeaKaE7wCWBnwkgI3UeMCNoN4Eycovo+YwoY20R7QXRRNnewt+2pjWA2ql6AbUSzrQMxWFfpupIXaU+9KFPY6rB3nU0lRlpBSWPnF5NgcIY6rDchecTKHcIafQcESalvqNuW/T8X9QSa/G+o9vO1L9JyvNwRaqhez/KKLTUnOhH9DZn8IzlSYBzDyjEuo9Yek26HlDvh9B+AfV34TqhktFJG64F1Z6UIjfh8iE3ReQ2XD7ktojchcuH3BWRe3D5kHsaidgJ6dyPb8DlwzdEzvfh8iHvi8gHcPmQD0TkAVw+5IGI/AYuH/IbEXkTLh/ypoi8DZcPeVtENuHyIZsich8uH3JfRG7B5UNuaWTxTJ3ANSI6fWFWXodyngdaigRqrovy3SDr6MPeCJjTnQKsPKvr8NePrQfoNC7AbgWMu6MCrDzytsFG+rGyLbpFq4kPe0vE7sAI8GN3ROzX6mUB9uuAmXZcgJXn2i6082Nl63sH7vzYOyL2LpT8WHmNugc1fuy9gBVjXIDdE7H31asCbIjVnxRgZbvfALvix8rrVBPa+7Eh1nRWgJXt6QF4MH6svFo9hFo/9qGIfaTeFGAfidjHYN392McBK+y7AqxZY8/SCtIjfySGGVtGrZXNSiyNgVpL4J9ka0tCvnEb6iVML8P0CDMQEdsZYjsQsZshdoPlSjM7mpK/K3NpZIhGIKKdrU1Ymortu1l7LCUBiHqGqC8hyjxSfNemL3PyLkyNhJxmKxeWQvo0yuw3lmI9Hsotr0HcyyF4bL+gkX+RoiWMoFBTZdReZGs8IyO6L0O8pujN9NLwkHHTzCq4qDciqu1BtUXUWw/qrYiaeVAzETX3oOYiys58F3cYMAKs/vFdLOiORwD7yMVXBF7BdVh1bsEcjWD87IEX+IBq7sHfBsXe0lUmGUbzuE5iluNpzhJPoLRQa1Bvo8I6xdcJzbAYJOOW93SMj3eY21joOcdW+CRbyaMsYxJOp0/y9DI66C1GNJ+q0blNNSfk3XGpGv5WNu9NqRp+izR+Ql48l6rhp1r66Slkb2ps8xTYBsymsda+LVelwfkXpmHKZ2nVRYuLb3WgxwzSe1OR/o5+MzuneC+bVGL92HI1GqnTvzTXvyo0rJ5TR8/VqKD3xF6vKUWVezLUca8tV5VhRKvoUMth76q+GWzT1W/GlKvR2AOPa5Ni7oVTrjp6x1lvbLkajQPFec8T8uRNuRqNHt2zPmy5Gg3MtrR0nG/LVS07aoBjZ1uuatWHlAXGHBCPea6xXtGE/KSZptYn/6A8W+P6/KvrGOZsnmUxQjkl69sW02lna1m5RMZfiMGqTSvKgf7FzPHB8jQWakOMr1iGaW59X6Vj13jU/C5oMYLZz3sAUs48AQlNTgKtdwIUL4lRV75nBrch4nCUHC2hDnXtVPQWLV/OGuXrnlOtFJfZ3lo9HpK9Tmnsjckn3CXNSnrYLXzDRRQlDe3mNCTTq6K7d3q+5rVfE3HjJcQ4G2kd2hHinbTyONWn9Yaj4/N6l2cKF+/52PGL2eYjbW0w5hmRLUJZyni67Uweya3DdfWisjlufhbRG0V7NSer0acdqVSMQk22mL3xBd1b2vu0J4c8mEYH3mOkqYwV75phFh3z6RFZVNfeSrxRXyZDx+WUrK6xx+XonoPuedDVY5xNWDHuQqkJMcM+3DUDopyzma5GpPGJ+izbHR3RGyyP6JOchTQ02N7EOQtZFmW/yFF5DWgcDRylh9NYpmPwhyuU5KjfJ4+NXfOW/zzt3Jr97RaN8eLRXJyJ6RLXDeIa0azhXV2+W+bAEiy8TzbIfy3vJfKrwhFtqMT1mcOZ9TKkHf+YItgxecYJzTZpduRbu/mp5SeG054ye+e4mz0iCxmR/YtgfRrRmIzoxz07YHbQ2SIkZCND7E4/8258vk5fHGPWj+srPtVgx1tMtmxG/A1dd3alNBY5YuB14GRpbBud7JIvGBPXibbudm6Xrz6ItOck3FHCFO1YuUD8P6Xf5seMk7WVEYEaxjeQalvnex8jillQRy1a5cttkGnrSvlJJsMzLbVd/6xMn+Qkq1PEhfLgat0Fzh26Z144SiYkd7rShtfRsmwuUh4v6RF7e0RRPNv9nl6BUe6LtEqu0Zw7pFHSg1EwzaII01bKIi/zLeeVpx5GO/2/ULe6zmsNKUbKZnBZQ1J+P6ZozZUygVHN4/eYZpNf65OlVuV8hjQWB85c/hZqP4bfRm5zH0annbMKN2gMMAV7ZzXCNdFKizBeN3K8zMg0tOy95WfHpGnl1pwmvmbrZmPseWUqezRq3uishSmfhsZLh8bLQB02aa/RatHUG0v0XIwtmnq3MpRfFW7NCpRnImXZIzOofoCUbiwVRrUrUpVjfIN6J9KqibRaMFvd3QB3zocg/XN9eXZ/m63ukbpJvk2HPDCOX7o0S/vkc5na8kiNKSDny9q+urP/kGqQe5ssKFLmc5w4Y3jXqUPXSSbpr/XKNiI7by2CObf0WrcxNvaQyp+vIAc0J1KalwZxmVrEWn5XjmjJIq07PkdEmf8W+VTsd5THzG5r+06inD9h402eVZYXRwpD0r+UedtZiV53nPg1ophwpr3rNtCq/oaRAmNMJsHvWab0hnCV450E9mjbZD9X7RTv4g0didZJ6oX6fYCN4ajXjnV3bJkem779Blqi1u1b97WQ+SXBHCV+p9nRa9GqNtA+6mLp/nS0WnqVy9+X6WG2xNfqY0Zt3MjCRnl5zKH6MpgLS1SNC2NCuFTrRRX5q0leRWbenQqlbFobyvlMA9uYFxQvSedAEeHz7i54vblPhX60V+i1CetS4xqJEmbjRjo/4FpazEpFSxGSWy+tSYmzHhWtF5aHu2pYO86WMiYrmCgpd8Ot3T4c5qIVORvDFDqKT/YWxYkuzS/hwt+R8kWJhmNIDrEBfu51tam23sOpiFe6zJnNiGrQJnSXYvCW7me+RbmOXjnUXfohHMJ59EHXkvR9WlGrys6UZcld6uH0X5M1mKhYlN62rN4Hl4vck1VOVfrTJwsn96avzDc5VftiOIT0JM8lnA/vb0i9OFLm26ZqfTDU5R7kOVThYc4zhL1z27o6L5dTub5WuYTy4HXA7LwYHO4AFscstl2IhZo4b+T9c0DrcFRC3awW/2s/DB/LqTqvUG4pfXP2MuCtc7tYZ2bRL64+Zyy3kNFczDGc5yjrnfWa/PzY/4sqvamR05v3Tx/9UjsGDK+F4nyoLB3j3VFk5Q2lgvsDPhlG6j/q72fkrxJeZTSK5KhCyexXFFMzLWRq5stLX+/MsxCZLJ0imfLUbDzRoJOxm2pH3YSfzcwDrHpKlL+p5L+I9X9H24XaI7IeJpvOGYRDqospC2J307p0b8/RFkmMZ3r5jG8TanBPfJdq8bzvXWqPZ36bub4Vf0nCc/2OGqluLjJZ3uWz86oNPcjvwHEuyHzvG9GZes5m8Qm0QcAeI5+j4kjJfP28IESX4sJlSReEMKOljHLbS7lNZ5LiAtrtXN86NMLHeqcf9x3wfH4ryy5F6rdU19KrA67UklR7HqmeUGagTfqvQYR2RV2Evxd12S/p3oqkKb2DvERvnGflJ8FOvOPCfs14nvJgJlM31+1GFNXb3cPyTGy9kAufeC/H90rwPUfKBr2tY4q7J6o8dzgroTnTMrn7uUNl8p6sB4xmW9n4KI+f5yW85gH9v12Ivu1Iug2ytCnbHtF+3oToJVo3WyQ9n6ssz9veKpHWfLXJNO3JSjsOzBnJ8j2BRI+74tnP5yClXE1cQMed63wiUzot0vdSkufnOOA0RCugt3JfQ3oqUZmJkswCvkSeB8gyD6BzJEhzJFLoiZJo+/D83Jr5Lz6i4sLBxvqlL9Yv399Y++qG/n9APlS/VL9SF2Dt+536Csb/ntoHTgP1R/UX9df6cf0P9T/V/8xNPzijMb9QuX/1v/0XbJNDCg==</latexit> Gene set <latexit sha1_base64="p9qBAtT0sQ9OxWrZqTMkFBJLPXA=">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</latexit> enrichment Geert-Jan Huizing Laura Cantini wk αk ATAC ATAC H αk RNA RNA H min (wk ),(Hm ) {∑ m ∑ k OT(Hm wk , αm k ) : wk ≥ 0,Hm ≥ 0}
  46. Trajectory Inference using Optimal Transport <latexit sha1_base64="dr9fl5ndkD+EfhnptSEBNz9eg14=">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</latexit> day 1 <latexit

    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day 18 <latexit 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[Waddington, 1936] Conrad Waddington vt vt+τ
  47. Trajectory Inference using Optimal Transport <latexit sha1_base64="dr9fl5ndkD+EfhnptSEBNz9eg14=">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</latexit> day 1 <latexit

    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day 18 <latexit 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[Waddington, 1936] Conrad Waddington Geoffrey Schiebinger vt vt+τ Gluing Optimal Transport: vt = ∇ψt inf vt {∥vt ∥2 : (Id + τvt )♯ αt = αt+τ}
  48. Gradient flows for genomics Gradient flows model: <latexit sha1_base64="LIz5cv8SkqGoJa7sBtLUbjFelcY=">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</latexit> ↵0

    αt+τ := arg min α ℰf (α) := 1 2τ W2 (αt , α)2 + f(α) Charlotte Bunne Geert-Jan Huizing Laura Cantini
  49. Gradient flows for genomics Gradient flows model: <latexit sha1_base64="LIz5cv8SkqGoJa7sBtLUbjFelcY=">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</latexit> ↵0

    αt+τ := arg min α ℰf (α) := 1 2τ W2 (αt , α)2 + f(α) Charlotte Bunne Learning from : f (αt , αt+τ ) inf f,α ℰf (α) − ℰf (αt ) Geert-Jan Huizing Laura Cantini f(α) = ∫ h(x)dα(x) h(x)
  50. ˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ

    e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + <latexit 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x1 <latexit 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x2 <latexit 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{xi }i Points cloud Positional encoding Token encoding Tokenize Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - Géométrie-Modélisation- Approximation) est issu de l ’A s s o c i a t i o n F r a n ç a i s e d ’A p p r o x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - G é o m é t r i e - M o d é l i s a t i o n - Approximation) est issu de l ’ A s s o c i a t i o n F r a n ç a i s e d ’ A p p ro x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. xi xj (Unmasked) Attention layer … next token probabilities Attention Norm MLP Classif N × …
  51. ˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ

    e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + <latexit 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x1 <latexit 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x2 <latexit sha1_base64="Fgw+vWgPriclgLxpVoHDEicBDLw=">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</latexit> {xi }i Points cloud Positional encoding Token encoding Tokenize Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - Géométrie-Modélisation- Approximation) est issu de l ’A s s o c i a t i o n F r a n ç a i s e d ’A p p r o x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - G é o m é t r i e - M o d é l i s a t i o n - Approximation) est issu de l ’ A s s o c i a t i o n F r a n ç a i s e d ’ A p p ro x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. xi xj (Unmasked) Attention layer Arbitrary number of tokens Arbitrary number of layers Expressivity Understanding … next token probabilities Attention Norm MLP Classif N × …
  52. Deep Transformers Tokens: X := (xi )i . Aω [X](x)

    := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Parameters: ω := (Q, K, V) x Aω [X](x)
  53. Deep Transformers Tokens: X := (xi )i . Aω [X](x)

    := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Parameters: ω := (Q, K, V) x Aω [X](x) Permutation invariance 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) α = 1 n ∑ k δxk
  54. Deep Transformers Tokens: X := (xi )i . Aω [X](x)

    := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Parameters: ω := (Q, K, V) Deep transformers with layers: T xi ↦ xi + 1 T 𝒜 ω [α](xi ) x Aω [X](x) Permutation invariance 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) α = 1 n ∑ k δxk
  55. Deep Transformers Tokens: X := (xi )i . Aω [X](x)

    := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Parameters: ω := (Q, K, V) Deep transformers with layers: T xi ↦ xi + 1 T 𝒜 ω [α](xi ) T → + ∞ dx(t) dt = 𝒜 ω [αt ](x(t)) x Aω [X](x) Permutation invariance 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) α = 1 n ∑ k δxk
  56. Deep Transformers Tokens: X := (xi )i . Aω [X](x)

    := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Parameters: ω := (Q, K, V) Deep transformers with layers: T xi ↦ xi + 1 T 𝒜 ω [α](xi ) T → + ∞ dx(t) dt = 𝒜 ω [αt ](x(t)) αt = 1 n ∑ i δxi (t) ∂αt ∂t + div( 𝒜 ω [αt ] αt ) = 0 Michael Sander x Aω [X](x) Permutation invariance 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) α = 1 n ∑ k δxk
  57. Deep Transformers Tokens: X := (xi )i . Aω [X](x)

    := ∑ j e⟨Qx,Kxj⟩Vxj ∑ ℓ e⟨Qx,Kxℓ⟩ Parameters: ω := (Q, K, V) Deep transformers with layers: T xi ↦ xi + 1 T 𝒜 ω [α](xi ) T → + ∞ dx(t) dt = 𝒜 ω [αt ](x(t)) αt = 1 n ∑ i δxi (t) ∂αt ∂t + div( 𝒜 ω [αt ] αt ) = 0 is not a Wasserstein flow! αt Twisted distance f(α) := ∬ e⟨Qx,Ky⟩dα(x)dα(y) minvt ,αt {∬ Sαt ∥vt ∥2dαt dt : div(αt vt ) + ∂t αt = 0} Sα (x) := ∫ e⟨Qx,Ky⟩dα(y) Michael Sander x Aω [X](x) Permutation invariance 𝒜 ω [α](x) := ∫ e⟨Qx,Ky⟩Vy dα(y) ∫ e⟨Qx,Kz⟩ dα(z) α = 1 n ∑ k δxk
  58. Gaussian Case and Clustering dμ dt + div(μΓθ [μ]) =

    0 Theorem: If , then μ(0) = 𝒩 (m(0), Σ(0)) · m = V(Id+ΣQ⊤K)m · Σ = VΣQ⊤KΣ + ΣK⊤QΣV⊤ Γθ [μ](x) := ∫ e⟨Qx,Ky⟩ ∫ e⟨Qx,Ky′  ⟩dμ(y′  ) Vy dμ(y) θ(t) = (Q(t), K(t), V(t)) μ(s) = 𝒩 (m(s), Σ(s)) t μ(0) μ(t) Valérie Castin
  59. Gaussian Case and Clustering dμ dt + div(μΓθ [μ]) =

    0 Theorem: If , then μ(0) = 𝒩 (m(0), Σ(0)) · m = V(Id+ΣQ⊤K)m · Σ = VΣQ⊤KΣ + ΣK⊤QΣV⊤ Γθ [μ](x) := ∫ e⟨Qx,Ky⟩ ∫ e⟨Qx,Ky′  ⟩dμ(y′  ) Vy dμ(y) θ(t) = (Q(t), K(t), V(t)) μ(s) = 𝒩 (m(s), Σ(s)) Theorem [Valérie Castin]: If and symmetric, stationary points of have rank less than V(t) = Id K(t)⊤Q(t) Σ(t) d/2. Conjecture: low-rank stationary covariances for any . K, Q, V … t μ(0) μ(∞) [Geshkovski, Letrouit, Polyanskiy, Rigollet 2023] The attention matrix converges to low-rank. → Clustering of for un-normalized attention. → μ t μ(0) μ(t) Valérie Castin
  60. Hugo Lavenant Sampling: diffusion transports are not optimal Costly to

    find … Straight flow easier to discretize! Conclusion
  61. Hugo Lavenant Sampling: diffusion transports are not optimal Costly to

    find … Straight flow easier to discretize! Optimizing: going deeper with ResNet Still a Wasserstein flow! Global convergence open … Raphaël Barboni Conclusion
  62. Hugo Lavenant Sampling: diffusion transports are not optimal Costly to

    find … Straight flow easier to discretize! Optimizing: going deeper with ResNet Still a Wasserstein flow! Global convergence open … Raphaël Barboni Conclusion Genomics: beyond linear functionals Scalability problems … Integrating multiomics … Geert-Jan Huizing <latexit 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Cost c <latexit 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RNA$RNA <latexit 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Space$ATAC time t + τ time t