A Kantorovich Monad for Ordered Spaces Paolo Perrone Tobias Fritz - PowerPoint PPT Presentation

A Kantorovich Monad for Ordered Spaces Paolo Perrone Tobias Fritz Max Planck Institute for Mathematics in the Sciences Leipzig, Germany FMCS 2018

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C X 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX PX X 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX • Unit δ : X → PX PX X 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. ? • Base category C 1/2 1/2 • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. ? ? • Base category C 1/2 1/2 • Functor X �→ PX 3/4 1/4 • Unit δ : X → PX 1/2 1/2 1/2 1/2 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. ? ? • Base category C 1/2 1/2 • Functor X �→ PX 3/4 1/4 • Unit δ : X → PX 1/2 1/2 1/2 1/2 • Composition E : PPX → PX [Lawvere, 1962] PX PPX 2 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a e : PA → A are “convex spaces” b A 3 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a a b e : PA → A are “convex spaces” a b b a b A PA 3 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a a b e : PA → A are “convex spaces” a b b a b A PA 3 of 20

Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a a b e : PA → A are λ a + (1 − λ ) b “convex spaces” a b • Formal averages are b mapped to actual a b averages A PA 3 of 20

Probability monads Radon monad [´ Swirszcz, 1974]: • Given a compact Hausdorff space X , PX is the set of Radon probability measures, equipped with the weak*-topology. 4 of 20

Probability monads Radon monad [´ Swirszcz, 1974]: • Given a compact Hausdorff space X , PX is the set of Radon probability measures, equipped with the weak*-topology. • Given a continuous map f : X → Y , Pf : PX → PY is given by push-forward: ( Pf )( p ) : A �→ p ( f − 1 ( A )) . 4 of 20

Probability monads Radon monad [´ Swirszcz, 1974]: • Given a compact Hausdorff space X , PX is the set of Radon probability measures, equipped with the weak*-topology. • Given a continuous map f : X → Y , Pf : PX → PY is given by push-forward: ( Pf )( p ) : A �→ p ( f − 1 ( A )) . • The map δ : X → PX assigns to x the Dirac delta measure δ x . 4 of 20

Probability monads Radon monad [´ Swirszcz, 1974]: • Given a compact Hausdorff space X , PX is the set of Radon probability measures, equipped with the weak*-topology. • Given a continuous map f : X → Y , Pf : PX → PY is given by push-forward: ( Pf )( p ) : A �→ p ( f − 1 ( A )) . • The map δ : X → PX assigns to x the Dirac delta measure δ x . • The map E : PPX → PX gives the average: � ( E µ ) : A �→ p ( A ) d µ ( p ) . PX 4 of 20

Probability monads Kantorovich monad [van Breugel, 2005, Fritz and Perrone, 2017]: • Given a complete metric space X , PX is the set of Radon probability measures of finite first moment, equipped with the Wasserstein distance , or Kantorovich-Rubinstein distance , or earth mover’s distance : � � � � � d PX ( p , q ) = sup f ( x ) d ( p − q )( x ) � � f : X → R � X � • Functorial and monad structure are analogous, where the morphisms are the short maps . 5 of 20

Probability monads Kantorovich monad [van Breugel, 2005, Fritz and Perrone, 2017]: • Given a complete metric space X , PX is the set of Radon probability measures of finite first moment, equipped with the Wasserstein distance , or Kantorovich-Rubinstein distance , or earth mover’s distance : � � � � � d PX ( p , q ) = sup f ( x ) d ( p − q )( x ) � � f : X → R � X � • Functorial and monad structure are analogous, where the morphisms are the short maps . • If X is compact, PX is compact [Villani, 2009]. 5 of 20

Probability monads Monad Category Algebras Compact convex subsets Radon KHaus of locally convex top. vector spaces a 6 of 20

Probability monads Monad Category Algebras Compact convex subsets Radon KHaus of locally convex top. vector spaces a Closed convex subsets Kantorovich CMet of Banach spaces b a [´ Swirszcz, 1974] b [Fritz and Perrone, 2017] 6 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . 7 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . • “Extending the y order by convexity” z x X 7 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . • “Extending the x y z y order by convexity” z x y z x x y z PX X 7 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . • “Extending the x y z y order by convexity” z x y z x x y z PX X 7 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . • “Extending the x y z y order by convexity” z x y z x y z x x y z PX X 7 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . • “Extending the x y z y order by convexity” z x y z x y z x y z x x y z PX X 7 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . • “Extending the x y z y order by convexity” z x y z x y z • “Moving the mass upwards” x y z x x y z PX X 7 of 20

The stochastic order Idea [Strassen, 1965] [Jones and Plotkin, 1989]: Constructing an order on PX entending the order on X . • “Extending the x y z y order by convexity” z x y z x y z • “Moving the mass upwards” x y z • “Larger measure to x x y z upper sets” PX X 7 of 20

The stochastic order Theorem [Strassen, 1965]: Let X be a Polish space equipped with a closed preorder. Let p , q be Radon probability measures on X . Then the following are equivalent: 8 of 20

The stochastic order Theorem [Strassen, 1965]: Let X be a Polish space equipped with a closed preorder. Let p , q be Radon probability measures on X . Then the following are equivalent: 1. For every closed upper set U , p ( U ) ≤ q ( U ); 8 of 20

The stochastic order Theorem [Strassen, 1965]: Let X be a Polish space equipped with a closed preorder. Let p , q be Radon probability measures on X . Then the following are equivalent: 1. For every closed upper set U , p ( U ) ≤ q ( U ); 2. There exists a coupling r ∈ P ( X × X ) of p and q entirely supported on { x ≤ y } . 8 of 20

The stochastic order Theorem [Strassen, 1965]: Let X be a Polish space equipped with a closed preorder. Let p , q be Radon probability measures on X . Then the following are equivalent: 1. For every closed upper set U , p ( U ) ≤ q ( U ); 2. There exists a coupling r ∈ P ( X × X ) of p and q entirely supported on { x ≤ y } . Definition: We say that p ≤ q in the usual stochastic order . 8 of 20

Ordered Wasserstein spaces Definition: Let X be a complete metric space with a closed preorder. We call Wasserstein space the space PX of Radon probability measures of finite first moment, equipped with the Wasserstein distance , or Kantorovich-Rubinstein distance , or earth mover’s distance : � � � � � d PX ( p , q ) = sup f ( x ) d ( p − q )( x ) � � f : X → R � � X and the usual stochastic order. 9 of 20