Gradient Flow in the Wasserstein Metric

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1 3 2 1 Time Position Graient Flow in the Wasserstein Metric Katy Craig University of California, Santa Barbara JMM, AMS Metric Geometry & Topology January 11th, 2018

2 graient flow an PDE Examples: t x(t) = r XE(x(t)) metric energy (L 2 (R ), k k L 2) E(f) = 1 Z rf 2 2 graient flow t f = f metric (P 2 (R ),W 2 ) energy E( ) = 1 2 Z K + Z V + 1 m 1 Z m graient flow = r ((rk ) )+r (rv )+ t m 2

3 Wasserstein metric Given two probability measures µ an on R, t : R! R transports µ onto if (B) =µ(t 1 (B)). Write this as t#µ =. The Wasserstein istance between μ an ν P2,ac(Rᵈ) is {z} effort to rearrange μ to look like ν, using t(x) {z} t sens μ to ν 3

4 geoesics Not just a metric space a geoesic metric space: there is a constant spee geoesic :[0, 1]! P 2 (R ) connecting any μ an ν. (0) = µ, (1) =, W 2 ( (t), (s)) = t s W 2 (µ, ) Monge Kantorovich µ Wasserstein geoesic (t) µ L 2 geoesic (1 t)µ + t [Peyré, Papaakis, Ouet 2013] 4

5 5 convexity Since the Wasserstein metric has geoesics, it has a notion of convexity. Recall: E: L 2 (Rᵈ) R is λ-convex if E((1 t)f + tg) apple (1 {z } t)e(f)+te(g) t(1 t) kf 2 gk 2 L 2 geoesic enpoints For any g L 2 (Rᵈ), E(f) =kf gk 2 is 2-convex =) 2 L 2 is NPC. Likewise, in the Wasserstein metric, E: P2(Rᵈ) R is λ-convex if E( (t)) apple (1 t)e(µ)+te( ) t(1 t) 2 W 2 2 (µ, ) W2 geoesic enpoints For any ν P2(Rᵈ), E(µ) =W is 2-concave =) 2 2 (µ, ) W2 is PC.

6 6 graient flow We want to efine the graient flow as, t (t) = r W 2 E( (t)) but without a Riemannian structure, we on t have a notion of graient. Given E: P2(Rᵈ) R, its local slope is: (E(µ) E( )) (µ) :=limsup!µ W 2 (µ, ) Given ρ:[0,t] P2(Rᵈ), its metric erivative is: 0 W 2 ( (s), (t)) (t) =lim s!t s t DEF: ρ(t):r P2(Rᵈ) is the Wasserstein graient flow of E:P2(Rᵈ) R if t E( (t)) apple 1 (t)) (t)

7 {z } 7 Wasserstein graient flow DEF: ρ(t):r P2(Rᵈ) is the Wasserstein graient flow of E:P2(Rᵈ) R if t E( (t)) apple 1 (t)) (t) Analogy with L 2 graient flow: Abbreviating r L 2 by r, t f(t) = re(f(t)) () f(t) = re(f(t)) t t E(f(t)) = re(f(t)) t f(t) () t E(f(t)) apple 1 2 re(f(t)) 1 2 t f(t)

8 {z} 8 graient flow an PDE t x(t) = r XE(x(t)) Goo news: graient flows structure is very useful in PDE existence uniqueness approximation {z } time iscretization contraction inequality stability Ba news: Wasserstein metric has more complicate geometry L 2 Wasserstein metric Riemannian manifol metric space non-positively curve positively curve

9 time iscretization: L 2 graient flow time iscretization f n f n 1 f(t) = re(f(t)), f(0) = g = re(f n ), f 0 = g t Analogous results hol in any NPC metric space [Mayer, 98], [CL 71] Define (h) = 1. 2 kh f n 1k E(h) What about when the metric space isn t NPC? Then f n solves r (f n )=0 convex () f n is the unique minimizer of Assume: E is λ-convex. Since L 2 (Rᵈ) is 1 NPC, is -convex. + Prop: kf n fn k 2 apple 1 1+ kf n 1 Thm: For = t, n kf(t) f nk 2 apple p C, n f n 1 k 2 kf(t) f(t)k2 apple e t kf(0) f(0)k2 time iscretization contraction inequality 9

10 10 time iscretization: W2 graient flow t E( (t)) apple 1 (t)) (t) time iscretization (JKO) n = arg min (0) = µ 0 = µ 1 2 W 2 2 (, n 1 )+E( ) Assume: E is boune below an λ-convex along generalize geoesics. 1 Then ( ) = 1 is -convex along gen geoesics. 2 W 2 2 (, n 1 )+E( ) + Thm: For = t, W 2 ( (t), n ) apple p C, W 2 ( (t), (t)) apple e t W 2 ( (0), (0)) [AGS 05] n n time iscretization contraction inequality 1 Prop: W 2 ( n, n ) apple 1+ W 2( n 1, n 1 )+O( 2 ) [C. 16] Overcome W2 geometry issues what about when E isn t λ-convex?

11 11 ω-convexity Recall: E: P2(Rᵈ) R is λ-convex if E( (t)) apple (1 t)e(µ)+te( ) t(1 t) 2 W 2 2 (µ, ) Def: Given a moulus of convexity ω(x) an λ R, E is ω-convex if E(( (t)) apple (1 t)e(µ)+te( ) 2 Examples:!(x) =x, reuces to λ-convexity!(x) =x log(x) (1 t)!(t 2 W 2 2 (µ, )) + t!((1 t) 2 W 2 2 (µ, )), [Ambrosio Serfaty, 2008] [Carrillo Lisini Mainini, 2014]!(x) =x p, p > 1, [Carrillo McCann Villani, 2006]

12 time iscretization: W2 graient flow t E( (t)) apple 1 (t)) (t) time iscretization (JKO) n = arg min (0) = µ 0 = µ 1 2 W 2 2 (, n 1 )+E( ) Assume: E is boune below an ω-convex Z along generalize geoesics 1 x for ω(x) satisfying Osgoo s conition:!(x) =+1 Thm: For = t, W 2 ( (t), n )! 0, [C. 17] n time iscretization 0 F 2t (W2 2 ( 1 (t), 2 (t))) apple W2 2 ( 1 (0), 2 (0)) t F t(x) =!(F t (x)) contraction inequality In particular, for ω(x) = x log(x) an W 2 ( (0), (0)) apple 1, W 2 ( (t), (t)) apple W 2 ( (0), (0)) e2 t 12

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From slow diffusion to a hard height constraint: characterizing congested aggregation

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