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 .
Monge problem
We are looking for an injective mapping with all measurable quantities that the functional
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.
Kantorovich problem
A relaxed problem was considered by Kantorovich in 1942. The Kantorovich problem looks for a probability measure on the product space with
for all compact quantities that the functional
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
for an injective mapping . In particular, the Monge problem also has a solution in this case.
Wasserstein metrics
For and probability measures on a metric space X let the set of all probability measures on with for all compact sets . Then defined
the pth water stone distance between and .
The pth Wasserstein distance defines a metric on the set of all probability measures .
If is a convex subset of and , then the geodesics of the pth Wasserstein metric are of the form
- ,
where the mapping defined by and the solution of the Kantorovich problem is to.
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 .
Generalizing this property, Lott and Villani and Sturm gave a synthetic definition of nonnegative Ricci curvature in general metric spaces.
swell
- ^ 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 .
- ^ 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 .
- ↑ 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 .