Optimal transportation

In mathematics , optimal transport describes a theory that has arisen from the analytical modeling of the transport problem. Lott and Villani as well as Sturm gave a synthetic definition of Ricci curvature bounds in general metric spaces with the help of optimal transport .

Optimal transport is originally a classic problem (going back to Monge and Kantorovich ) which, based on a given initial distribution and a desired final distribution, searches for the cheapest transport in which the initial distribution is converted into the final distribution.

The initial and final distributions are modeled by density functions ( probability measures ) and on metric spaces and . The cost function is a given function . The value indicates the cost of transport from to . A typical example is if and are subsets of a normalized vector space , or more generally for a differentiable function . ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle c:X\times Y\rightarrow \mathbb {R} }$ ${\displaystyle c(x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle c(x,y)=\|x-y\|}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle c(x,y)=h(x-y)}$ ${\displaystyle h}$

Monge problem

We are looking for an injective mapping with all measurable quantities that the functional ${\displaystyle r:X\rightarrow Y}$ ${\displaystyle \mu (r^{-1}(B))=\nu (B)}$ ${\displaystyle B\subset Y}$

${\displaystyle \int _{X}c(x,r(x))d\mu (x)}$

minimized.

There are examples where the Monge problem has no solution, e.g. B. if is a Dirac measure and the sum of at least two Dirac measures. ${\displaystyle \mu }$ ${\displaystyle \nu }$

Kantorovich problem

A relaxed problem was considered by Kantorovich in 1942. The Kantorovich problem looks for a probability measure on the product space with ${\displaystyle \pi }$ ${\displaystyle X\times Y}$

${\displaystyle \pi (A\times Y)=\mu (A),\pi (X\times B)=\nu (B)}$

for all compact quantities that the functional ${\displaystyle A\subset X,B\subset Y}$

${\displaystyle \int _{X\times Y}c(x,y)d\pi (x,y)}$

minimized.

Kantorovich proved that such a probability measure always exists.

If and for a strictly convex function h then the solution of the Kantorovich problem is of the form ${\displaystyle X=Y=\mathbb {R} ^{n}}$ ${\displaystyle c(x,y)=h(x-y)}$

${\displaystyle \pi =(id\times r)_{*}\mu }$

for an injective mapping . In particular, the Monge problem also has a solution in this case. ${\displaystyle r:X\rightarrow Y}$

Wasserstein metrics

→ Main article : Wasserstein metric

For and probability measures on a metric space X let the set of all probability measures on with for all compact sets . Then defined ${\displaystyle p\geq 1}$ ${\displaystyle \mu ,\nu }$ ${\displaystyle \Gamma (\mu ,\nu )}$ ${\displaystyle X\times X}$ ${\displaystyle \pi (A\times X)=\mu (A),\pi (X\times B)=\nu (B)}$ ${\displaystyle A,B\subset X}$

${\displaystyle W_{p}(\mu ,\nu ):=\left(\inf _{\gamma \in \Gamma (\mu ,\nu )}\int _{X\times X}d(x,y)^{p}\,\mathrm {d} \gamma (x,y)\right)^{1/p}}$

the pth water stone distance between and . ${\displaystyle \mu }$ ${\displaystyle \nu }$

The pth Wasserstein distance defines a metric on the set of all probability measures . ${\displaystyle W_{p}}$ ${\displaystyle X}$

If is a convex subset of and , then the geodesics of the pth Wasserstein metric are of the form ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle p>1}$

${\displaystyle t\rightarrow (F_{t})_{*}\pi }$,

where the mapping defined by and the solution of the Kantorovich problem is to. ${\displaystyle F_{t}:X\times X\rightarrow X}$ ${\displaystyle F_{t}(x,y)=tx+(1-t)y}$ ${\displaystyle \pi }$ ${\displaystyle c(x,y)=d(x,y)^{p}}$

Ricci curvature bounds

Let M be a compact Riemann manifold with the probability measure given by the volume form. Then M has nonnegative Ricci curvature if and only if there is a connecting geodesic (with respect to the W ₂ -Wasserstein metric) for every two probability measures , along which the entropy functional is convex . ${\displaystyle \mu ,\nu }$

Generalizing this property, Lott and Villani and Sturm gave a synthetic definition of nonnegative Ricci curvature in general metric spaces.

swell

1. ^ John Lott , Cédric Villani : Ricci curvature for metric-measure spaces via optimal transport. In: Annals of Mathematics . Vol. 169, 2009, pp. 903-991, (PDF; 552 kB), doi : 10.4007 / annals.2009.169.903 .
2. ^ Karl-Theodor Sturm: On the geometry of metric measure spaces. ( Memento of the original from June 28, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. In: Acta Mathematica . Vol. 196, No. 1, 2006, 65-131, (PDF; 591 kB), doi : 10.1007 / s11511-006-0002-8 . 
3. ↑ Wilfrid Gangbo, Robert J. McCann: The geometry of optimal transportation. In: Acta Mathematica. Vol. 177, No. 2, 1996, 113-161, (PDF; 2.8 MB), doi : 10.1007 / BF02392620 .