Gabriel Peyré É C O L E N O R M A L E S U P É R I E U R E Optimal and Diffusion Transports in Machine Learning to be or not to be a gradient ∇

Generative AI: auto-regressive vs. transport Raconte de façon rigolotte l'histoire d'un chercheur CNRS qui présente l’IA générative devant un large auditoire. Le professeur Jacques Leclair, célèbre directeur de l’Institut de Mathématiques du CNRS, monte sur scène avec l’enthousiasme d’un enfant qui vient de découvrir que son yaourt préféré a un nouveau goût. Devant lui, un large auditoire aux regards curieux. “Aujourd’hui, mes amis,” commence-t-il avec un clin d’œil complice, “je vais vous prouver que les mathématiques peuvent être aussi cool qu’un chaton jouant avec une pelote de laine… et je parle bien sûr de l’IA générative!”. Représente un chercheur en mathématiques en train de présenter l’IA générative devant un large auditoire. DALL·E 2

Generative AI: auto-regressive vs. transport Raconte de façon rigolotte l'histoire d'un chercheur CNRS qui présente l’IA générative devant un large auditoire. Le professeur Jacques Leclair, célèbre directeur de l’Institut de Mathématiques du CNRS, monte sur scène avec l’enthousiasme d’un enfant qui vient de découvrir que son yaourt préféré a un nouveau goût. Devant lui, un large auditoire aux regards curieux. “Aujourd’hui, mes amis,” commence-t-il avec un clin d’œil complice, “je vais vous prouver que les mathématiques peuvent être aussi cool qu’un chaton jouant avec une pelote de laine… et je parle bien sûr de l’IA générative!”. Représente un chercheur en mathématiques en train de présenter l’IA générative devant un large auditoire. DALL·E 2 Le professeur Jacques Leclair, célèbre directeur de l’Institut de Mathématiques du CNRS monte Pre-training: next token prediction. Generation: auto-regressive. Pre-training: denoising. Generation: dynamic transport.

Sampling by Transportation

Sampling by Transportation

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

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

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

Sampling Optimizing Transformers Genomics

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 ?

Gradient Structure Riemannian structure at : α ∥v∥2 L2(α) := ∫ ℝd ∥v(x)∥2dα(x) Otto’s Calculus Density manifold Felix Otto John Lafferty 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) Eulerian αt Lagrangian vt ?

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 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) Eulerian αt Lagrangian vt ?

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

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

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

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 model) 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

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 )

min vt {∫ ∥vt ∥2 L2(αt ) dt : div(αt vt ) + ∂t αt = 0} Diffusion model: α0 αt α1 Diffusion vs Optimal Transport Optimal transport: min vt ,αt {∫ ∥vt ∥2 L2(αt ) dt : div(αt vt ) + ∂t αt = 0} =: W2 (α0 , α1 )2 (Wasserstein distance) T αt := Law((1 − t)X0 +tX1 )

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 Optimal transport: min vt ,αt {∫ ∥vt ∥2 L2(αt ) dt : div(αt vt ) + ∂t αt = 0} =: W2 (α0 , α1 )2 (Wasserstein distance) T αt := Law((1 − t)X0 +tX1 )

Sampling Optimizing Transformers Genomics

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)

αt+τ := arg min α 1 2τ W2 (αt , α)2 + f(α) Implicit Euler step: 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 Optimization over distributions: min α f(α)

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)

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)

Training Dynamics of 2 layers MLP 2 layers perceptron: ϕx (z) = u σ(⟨z, v⟩) σ (uk )k (vk )k z g(x1 ,…,xn ) (z) g(x1 ,…,xn ) (z) := 1 n ∑n k=1 ϕxk (z) parameters , x = (u, v)

min x1 ,…,xn F(x1 , …, xn ) := ∑ i ∥gx1 ,…,xn (zi ) − yi ∥2 Training dxi (t) dt = − ∇xi F(xi (t)) Training Dynamics of 2 layers MLP 2 layers perceptron: ϕx (z) = u σ(⟨z, v⟩) σ (uk )k (vk )k z g(x1 ,…,xn ) (z) g(x1 ,…,xn ) (z) := 1 n ∑n k=1 ϕxk (z) parameters , x = (u, v)

min x1 ,…,xn F(x1 , …, xn ) := ∑ i ∥gx1 ,…,xn (zi ) − yi ∥2 Training dxi (t) dt = − ∇xi F(xi (t)) Training Dynamics of 2 layers MLP f(α) = ∫ kdα ⊗ α + ∫ hdα k(θ, θ′  ) := ∑ i ⟨ϕθ (zi )ϕθ′  (zi )⟩ h(θ) := − ∑ i ⟨yi , ϕθ (zi )⟩ α = 1 n ∑ k δxk minα f(α) := ∑ i ∥Gα (zi ) − yi ∥2 ∂αt ∂t − div(∇W f(αt ) αt ) = 0 Gα (z) := ∫ ϕθ (z)dα(θ) 2 layers perceptron: ϕx (z) = u σ(⟨z, v⟩) σ (uk )k (vk )k z g(x1 ,…,xn ) (z) g(x1 ,…,xn ) (z) := 1 n ∑n k=1 ϕxk (z) parameters , x = (u, v)

min x1 ,…,xn F(x1 , …, xn ) := ∑ i ∥gx1 ,…,xn (zi ) − yi ∥2 Training dxi (t) dt = − ∇xi F(xi (t)) Training Dynamics of 2 layers MLP f(α) = ∫ kdα ⊗ α + ∫ hdα k(θ, θ′  ) := ∑ i ⟨ϕθ (zi )ϕθ′  (zi )⟩ h(θ) := − ∑ i ⟨yi , ϕθ (zi )⟩ α = 1 n ∑ k δxk minα f(α) := ∑ i ∥Gα (zi ) − yi ∥2 ∂αt ∂t − div(∇W f(αt ) αt ) = 0 Gα (z) := ∫ ϕθ (z)dα(θ) 2 layers perceptron: ϕx (z) = u σ(⟨z, v⟩) 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 g(x1 ,…,xn ) (z) g(x1 ,…,xn ) (z) := 1 n ∑n k=1 ϕxk (z) parameters , x = (u, v)

Sampling Optimizing Transformers Genomics

˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + 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 x1 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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 × …

˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + 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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 Arbitrary number of tokens Arbitrary number of layers Expressivity Understanding … next token probabilities Attention Norm MLP Classif N × …

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)

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

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

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

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 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

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

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

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

. . . . . . . . . on . . . . . . . . . . . . . . . . OT as a cell-cell distance . . . . . . . . . . . . . . . . . . Wasserstein Singular Vectors . . . . . . . . . . . . . . . . . . OT-based joint dimensionality reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OT-based trajectory inference lications of multimodal omics CD103+ CD8+ TEM Zebrafish embryos along development Sampling Optimizing Transformers Genomics

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

Single Cell Multi-omics Understanding cell diversity: many types, many states for each type. Applications: cancer mutations, dynamic of adaptation, development, … 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2 Rd Tissue Dissociation isolation RNA amplification sequencing … 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 d ⇠ 102

Single Cell Multi-omics Understanding cell diversity: many types, many states for each type. Applications: cancer mutations, dynamic of adaptation, development, … 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2 Rd Tissue Dissociation isolation RNA amplification sequencing … 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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 AABDDXictVzbchu5EYU3t7VzWW/ymJdJZKe8KUeRFFeSqq1UrXWxrLVsyyYle3dpu3gZ0rSHHJpDyheuviEfkNfkE/KWSh7zDfmD5CkP+YH0BRhgSMw0RnE8ZQkD4nQ3GkCjuwGqM0mG2Wxj4x8XPvjGN7/17e98ePHSd7/3/R98dPnjH55k6XzajY+7aZJOH3faWZwMx/HxbDhL4seTadwedZL4UeflDn7+6DSeZsN03Jy9ncRPRu3BeNgfdtszqHp2+aMrvaiVDUfR5sbTxY2zK88ur22sb9C/aLWwqQtrSv87Sj+O/qNaqqdS1VVzNVKxGqsZlBPVVhk8X6lNtaEmUPdELaBuCqUhfR6rM3UJsHNoFUOLNtS+hJ8DePtK147hHWlmhO4ClwT+TwEZqauASaHdFMrILaLP50QZa8toL4gmyvYWfnc0rRHUztRzqJVwpmUoDvsyU331W+rDEPo0oRrsXVdTmZNWUPLI6dUMKEygDss9+HwK5S4hjZ4jwmTUd9Rtmz7/J7XEWnzv6rZz9S+S8io8kWro3qc5hbY6JfoRjeYcPmN5EuA8AAqx7iOWXpOuR9T7MbRfQP09eM6oZHTSgWdBtWeVyB14fMgdEbkPjw+5LyIP4fEhD0XkETw+5JFGInZKOvfjG/D48A2R8wN4fMgHIvIhPD7kQxF5Ao8PeSIiv4THh/xSRN6Cx4e8JSLvwOND3hGRTXh8yKaIPIbHhzwWkXvw+JB7Glm+UqfwpERnKKzKm1Au8kBLkUDNTVG+bbKOPux2wJrulmDlVb0Lv/3Y3QCdxiXYvYB51y/ByjNvH2ykHyvbotu0m/iwt0XsAcwAP/ZAxH6uXpRgPw9YaS9LsPJaO4R2fqxsfe/Cmx97V8Teg5IfK+9R96HGj70fsGNMSrBHIvaBelWCDbH60xKsbPcbYFf8WHmfakJ7PzbEms5LsLI9PQEPxo+Vd6tHUOvHPhKxj9WbEuxjEfsFWHc/9ouAHfZdCdbssZdoBxmQPxLDiq2i1s5XJZYmQK0t8E/yvSUh37gD9RJmkGMGhBmJiP0csR+IOMwRh8FyZbkdzcjflbk0ckQjENHJ9yYszcT2vbw9lpIAxG6O2F1CVHmkONamL6fkXZgaCTnLdy4shfQpze03lmI9H6otr0HcLyB4bj+nmX+doiWMoFBTVdSe53s8IyN6r0K8pujN9NLwkHGz3Cq4qDciquNBdUTUWw/qrYiae1BzEXXqQZ2KKLvyXVwrYAZY/eNYLOiNZwD7yOVPBF7BTdh1bsMajWD+HIEX+JBq7sPvBsXe0lMlGUbzuE9iluNJwRJPobRQa1Bvo8Jdiq8TWmExSMYt7+sYH98wt7HQa46t8Fm+k0d5xiSczpDkGeR00FuMaD3Vo3OHas7Iu+NSPfztfN2bUj38Hmn8jLx4LtXDz7T0s3PI3tTY5jmwDVhNE619W65Lg/MvTMOUL9GuixYXR3Wk5wzSe1OT/oEemYNzjMsOlVg/tlyPRub0Lyv0rw4Nq+fM0XM9Kug9sddrSlHtnox13GvLdWVIaRcdaznsW92RwTY9PTKmXI/GEXhcOxRzL5xy3dk7yXtjy/VonCjOe56RJ2/K9WgM6J31Ycv1aGC2pa3jfFuua9lRAxw723Jdqz6mLDDmgHjOc431iqbkJ801tSH5B9XZGtfnX93HMGfzNI8RqilZ37acTiffy6olMv5CDFZtVlMO9C/mjg9WpLFQW2J8xTLMCvv7Kh27x6PmD0GLEax+PgOQcuYJSGhyEmi9E6C4KUZdxZ4Z3JaIw1nSX0K1dO1M9BYtX84aFeueUa0Ul9neWj22yF5nNPcm5BMekmYlPRyWjnAZRUlDhwUNyfTq6O6dXq9F7W+IuMkSYpLPtC6dCPFJWnWc6tN6w9HxVX3KM4OHz3zs/MVsc19bG4x5UrJFKEsVT7edySO5dbivXlc2x82fRTSiaK9OyWoM6UQqE6NQky1mb3xB75b2MZ3JIQ+m0YVxjDSVieJTM8yiYz49Iovq2luJN+rLZOi4nJHVNfa4Gj1w0AMPun6MswM7xj0oNSFmOIa3ZkCUcynXVUoan6pf5KejKY1gdUSfFCykocH2Ji5YyKoo+3mBymtA42zgKD2cxjIdg2+tUJKjfp88NnYtWv6rdHJrzrfbNMfLZ3N5JqZHXLeIa0Srhk91+W2ZA0uw8H6yRf5rdS+RXx2OaEMlrk8dzqyXMZ34xxTBTsgzTmi1Sauj2NrNTy1/YjgdKXN2jqfZKVnIiOxfBPtTSnMyov/u3QFzgs4WISEbGWJ3hrl34/N1huIcs37cUPGtBjvfYrJlc+Jv6LqrK6O5yBED7wNnS3Pb6OSQfMGYuE61dbdru3r3QaS9J+HOEqZo58o14v8J/TT/zTxZW5kRqGEcgUzbOt94pBSzoI7atMtX2yDT1pXySi7DUy213f+sTFcKku1SxIXy4G7dA85demdeOEumJHe20ob30apsLlKeLOkRe9unKJ7t/kDvwCj3ddol12jNtWiWDGAWzPIowrSVssjLfKt5FamH0c7+L9StrotaQ4qRshlc1pCU348pWnOlTGBW8/x9SavJr/XpUqtqPmOaiyNnLX8NtT+Bn0Zu8x5Gp1OwCts0B5iCfbMa4ZpopUUYr+0CLzMzDS37bvnZOWlauTXnia/ZutkY+7Q2lSOaNW901sKUz0PjhUPjRaAOm3TWaLVo6o0leibGFk19WhnKrw63Zg3Kc5Gy7JEZ1DBASjeWCqPaE6nKMb5BvRNpbYi02rBa3dMAd82HIP1rfXl1f53v7pG6Rb5Nlzwwjl96tEqH5HOZ2upIjSkg5xvavrqrv0U1yL1DFhQp8z1OXDF86tSl5yyX9Gd6Z0vJzluLYO4tvdZtjI1tUflXK8gRrYmM1qVB3KAWsZbflSNaskjrjs8RUea/TT4V+x3VMbPb2o5JVPAnbLzJq8ry4khhTPqXMm8HK9HrgRO/RhQTzrV33QFa9UcYKTDGZBL8nmVGI4S7HJ8ksEfbIfu5aqf4FG/sSLROUi/U7wJsDEe9dq67c8v02PTt59AStW5H3ddC5pcEc5T4nedEr0272kj7qIul9/PRautdrvhepYf5El+rjzm1cSMLG+UVMS31aTAXlqgeF8aEcKnXizry15O8jsx8OhVK2bQ2lIuZBrYxzyleku6BIsLn3V3zenOfCP3orNDrENalxjUSJczGpTo/4FpazEpdXNmHuPZi5W6UODtR2U5hqLu7hbXfbCFjsn6JknI23NqVvVWIUuQsDFPoKr7RWxYfujQ/hQd/RsoXHRqOIbnDBvi3N9WO2nsPtyFe6TJnNCOqQVvQW4q927qfxRbVOnrlUHfph3AI5zEEXUvSD2knrSs7U5Yld6mH039NVmCqYlF627J+H1wuck9WOdXpz5Asm9yboTLfxanbF8MhpCdFLuF8+FxD6kVfme801euDoS73oMihDg9zjyFszG3r+rxcTtX6WuUSyoN3AXPiYnB48lceq9h2IRZq6ozI++eA1qFfQd3sFv9rPwwfy6k+r1BuGX3X7EXAqHO7WGdk0R+uv2Yst5DZXM4xnGea9856S35+7PdFtUYqdXrz/umjP2rngOG1UJwHlaVjvDuLrLyhVPBcwCdDqv6t/nZB/jbCq5xGmRx1KJlzinJqpoVMzXzj0tc781mITJZOmUxFajaOaNCN2B11oG7B/53cA6x7O5S/S8m/Eev//mwPavtkPUwWnTMHLaqLKfthT9F69G7vz5ZJjHd5+W5vE2rwLPyQavGe7z1qj3d9m4W+lX+DhNf6XZWqXiEiWT7ds+uqAz0onrxxDsh8zzeiu/ScxeKbZ6OAs0Vzf2pZogV9It8s6JTiO46UXZqrE31WjycHeMO+neeHIvVLqmtrO497rsT5qJTz0RLnjLRT5PDG+az6blYZlx2HSy/PnZ3qdinF2fY8rzo3ulvKhe+gV+MHFfiBI2WDtP+SIuGpqs7mzStozrVM7gnrWJlMJOsB48x2Pt7Vke1pBa/TgP7fKUXfcSTdB1k6lP+O6IRtSvQSrZs9kp5vOlZnUm9XSGu+R8k07V1HOw/MrcXqLH2i5x1nLsxfI1jQesDd1NxMlLIncQmdDt0IjIkS35GsptQX5OmLFAaiJHqmPru8trn8dyBWCydb65u/Xr/xYGvts239NyI+VD9WP1XXwD7+Rn0GI3GkjhX+rYg/qD+qP23/fvvP23/Z/is3/eCCxvxIFf5t//2/6U7mBQ== d ⇠ 104 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d ⇠ 105

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

Optimal Transport Across Cells 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day 1 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day 18 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[Waddington, 1936] Conrad Waddington Geoffrey Schiebinger vt vt+τ

Optimal Transport Across Cells 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day 1 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day 18 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[Waddington, 1936] Conrad Waddington Geoffrey Schiebinger vt vt+τ Gluing Optimal Transport: vt = ∇φt

Gradient flows for genomics Better modeling the dynamics: use gradient flows 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 ↵0 αt+τ := arg min α 1 2τ W2 (αt , α)2 + f(α) Charlotte Bunne

Gradient flows for genomics Better modeling the dynamics: use gradient flows 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 ↵0 Geert-Jan Huizing Laura Cantini αt+τ := arg min α 1 2τ W2 (αt , α)2 + f(α) 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 ATAC 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Cost c 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Potential f Key issues: learning the potential f(α) Integrate several omics time t + τ time t Charlotte Bunne

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

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

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 Transformers: understanding in-context learning Transformers are universal! Nothing else is known … Takashi Furuya Conclusion