¸ = 100 Matching Iterations of the semi-discrete OT algorithm minimizing (5.8) (here dient descent is used). The support (yj ) j of the discrete measure — is the red points, while the continuous measure – is the uniform measure The blue cells display the Laguerre partition (Lg(¸) (yj )) j where g ( ¸ ) is dual potential computed at iteration ¸. wton solver which is applied to sampling in computer graph- sed in [De Goes et al., 2012], see also [Lévy, 2015] for appli- -D volume and surface processing. An important area of ap- semi-discrete method is for the resolution of incompressible mic (Euler’s equations) using Lagrangian methods de Goes ], Gallouët and Mérigot [2017]. The semi-discrete OT solver ompressibility at each iteration by imposing that the (possi- d) points cloud approximates a uniform inside the domain. rgence (with linear rate) of damped Newton iterations is f (a, b) and s igure 6.1: Example of computation of W1(a, b) on a planar graph with uniform weights wi,j = 1. eft: potential f solution of (6.5) (increasing value from red to blue). The green color of the edges is oportional to |(Òf) i,j |. Right: flow s solution of (6.6), where bold black edges display non-zero si,j , hich saturate to wi,j = 1. These saturating flow edge on the right match the light green edge on the ft where |(Òf) i,j | = 1. 1. Algorithmic Foundations 2. Entropic Regularization 3. (Gradient) Flows in ML (to be or not to be a gradient field)
y3 y4 y5 min 2⌃n n X i=1 |xi y (i) |p, p > 1 <latexit 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<latexit 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image color statistics Y Source image after color tra Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Sliced Wasserstein projection image color statistics Y Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Style image (Y) Sliced Wasserstein projection of X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization optimal In 3-D: Color Image Palette Equalization Reference <latexit 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sha1_base64="3GKcotffcbQ7ZK6RRi5K3JhAuCs=">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</latexit> Output <latexit 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<latexit 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<latexit 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<latexit 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Input <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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transport <latexit 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9 Discrete d = 1 Discrete d = 2 Density d = 1 Density d = 2 Figure 2.1: Schematic display of discrete distributions – = q n i =1 ai”xi (red cor- responds to empirical uniform distribution ai = 1/n, and blue to arbitrary distri- butions) and densities d–(x) = fl– (x)dx (in violet), in both 1-D and 2-D. Discrete distributions in 1-D are displayed using vertical segments (with length equal to ai ) and in 2-D using point clouds (radius equal to ai ). is the dimension), can have a density d–(x) = fl – (x)dx w.r.t. the Lebesgue measure, often denoted fl – = d – d x , which means that ⁄ ⁄ ↵ = P i ai xi Positive Radon measure ↵ on a metric space X. X = Rd <latexit 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9 Discrete d = 1 Discrete d = 2 Density d = 1 Density d = 2 Figure 2.1: Schematic display of discrete distributions – = q n i =1 ai”xi (red cor- responds to empirical uniform distribution ai = 1/n, and blue to arbitrary distri- butions) and densities d–(x) = fl– (x)dx (in violet), in both 1-D and 2-D. Discrete distributions in 1-D are displayed using vertical segments (with length equal to ai ) and in 2-D using point clouds (radius equal to ai ). is the dimension), can have a density d–(x) = fl – (x)dx w.r.t. the Lebesgue measure, often denoted fl – = d – d x , which means that ⁄ ⁄ ↵ = P i ai xi Positive Radon measure ↵ on a metric space X. X = Rd <latexit 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Probability (normalized) measure: ↵(X) = R X d↵(x) = 1 Integration against continuous functions: Measure of sets A ⇢ X: ↵(A) = R A d↵(x) > 0 R X g(x)d↵(x) > 0 d↵(x) = ⇢↵(x)dx ↵ = P i ai xi R X gd↵ = P i aig(xi)
will make a detour through a more general linear optimization problem of the f for some compact convex set C. We first introduce the notion of extremal points, which are vertices of C Extr(C) := P ; ↑ (Q, R) → C 2, P = Q + R 2 ↘ Q = R . So to show that P / → Extr(C) is su!ces to split P as P = Q+R 2 with Q ≃= R and (Q, R) → C2. the following fundamental result. Proposition 4. If C is compact, then Extr(C) ≃= 0. The fact that C is compact is crucial, for instance, the set (x, y) → R2 + ; xy ↫ 1 has no We can now use this result to show the following fundamental result, namely that th solution to a linear program which is an extremal point. Note that of course, the set of solu non-empty because one minimizes a continuous function on a compact) might not be a sing 11 Figure 1: Left: extremal points of a convex set. Right: the solution of a convex program Proposition 5. If C is compact, then Extr(C) → argmin P→C ↑C, P↓ ↔= ↗. Proof. One consider S := argmin P→C ↑C, P↓. We first note that S is convex (as always fo compact because C is compact and the objective function is continuous so that Extr(S) that Extr(S) ↘ Extr(C). [ToDo: finish] al points of a convex set. Right: the solution of a convex program is a convex set. compact, then Extr(C) → argmin P→C ↑C, P↓ ↔= ↗. := argmin P→C ↑C, P↓. We first note that S is convex (as always for an argmin) and ompact and the objective function is continuous so that Extr(S) ↔= ↗. We will show . [ToDo: finish] em states that the extremal points of bistochastic matrices are the permutation 𝒞 C Argmin Figure 1: Left: extremal points of a convex set. Right: the solut Proposition 5. If C is compact, then Extr(C) → argmin P→C ↑C, P↓ Proof. One consider S := argmin P→C ↑C, P↓. We first note that S Figure 1: Left: extremal points of a convex set. Right: the solution of a convex pro Proposition 5. If C is compact, then Extr(C) → argmin P→C ↑C, P↓ ↔= ↗. Proof. One consider S := argmin P→C ↑C, P↓. We first note that S is convex (as alway compact because C is compact and the objective function is continuous so that Extr( that Extr(S) ↘ Extr(C). [ToDo: finish] The following theorem states that the extremal points of bistochastic matrices matrices. It implies as a corollary that the cost of Monge and Kantorovitch are the share a common solution. Theorem 2 (Birkho! and von Neumann). One has Extr(Bn ) = Pn . Proof. We first show the simplest inclusion Pn ↘ Extr(Bn ). Indeed it follows from the
x g✓ ⇢ ⇢ z x Discriminative Generative z1 z2 g✓ Z x z X d⇠ d⇠ ⇠1 ⇠2 g✓(z) = ⇢(✓K(. . . ⇢(✓2(⇢(✓1(z) . . .) Deep networks: d⇠(x) = ⇢(⇠K(. . . ⇢(⇠2(⇢(⇠1(x) . . .)