Abstract
In this chapter we use the minimal value of transport problems between two probabilities in order to define a distance on the space of probabilities. We mainly consider costs of the form \(c(x,y) = \vert x - y\vert ^{p}\) in \(\varOmega \subset \mathbb{R}^{d}\). We analyze the properties of the distance (called Wasserstein distance) that it defines, in connection with the weak convergence. We study then curves in the space of probability endowed with this distance, proving that their behavior is ruled by the continuity equation \(\partial _{t}\mu _{t} + \nabla \cdot (\mathbf{v}_{t}\mu _{t}) = 0\) for L p vector fields v t . The geodesics of this space are also studied. In the discussion section, these distances are compared to other important distances on measures, and the notion of barycenter of a finite number of measures is introduced.
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Notes
- 1.
The name is highly debated, in particular in Russia, since L. Vaserstein (whose name is sometimes spelled Wasserstein), a Russian scholar based in the United States, did not really play the key role that one could imagine in the introduction of these distances. Yet, this is nowadays the standard name in Western countries, probably due to the terminology used in [198, 246], and it seems impossible to change this convention, even if other names have been often suggested, such as Monge-Kantorovich distances, etc. Also note than in the most applied communities the name Earth Mover Distance (EMD) is very common for this distance.
- 2.
In particular in image processing. Thanks to G. Peyré for pointing out this interpretation.
- 3.
It is not strictly necessary to use absolutely continuous measures: we actually need atomless measures, and then we can apply Theorem 1.33.
- 4.
The existence of cutoff functions, i.e., a sequence of continuous compactly supported functions converging pointwisely to 1, is indeed peculiar to the case of locally compact spaces.
- 5.
The proof that we gave of the second part is standard and based on the interpretation of the continuity equation that we gave in Chapter 4 The approximations that we performed are the same as in [15], but we tried to simplify them to make this exposition self-contained: in order to do that, we chose a very precise convolution kernel. Concerning the first part, we stress that a different, and much more elegant, approach can be found in [15] and does not use approximation. The approach that we presented here is more or less inspired by a paper by Lisini, [212].
- 6.
One of the difficulties in this result is that the functions \(t\mapsto \int \varphi _{t_{0}}\varrho _{t}^{(1)}\) and \(t\mapsto \int \psi _{t_{0}}\varrho _{t}^{(2)}\) are differentiable for a.e. t, but not necessarily at t = t 0 (we know that for every integrand the integral is differentiable for a.e. T, and not that for a.e. t differentiability occurs for every integrand). We propose a way to overcome this difficulty via Lipschitz solutions. As an alternative approach, one can try to find the precise set of times t such that differentiability occurs for every integrand, as in [155], based on ideas from [15].
- 7.
Indeed, one can define sliced Wasserstein distances associated with other exponents p, and [65] proves \(W_{1} \leq C\mathrm{S}W_{1}^{1/(d+1)}\); then, one can compose with W 2 ≤ CW 1 1∕2 to obtain the result for p = 2.
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Santambrogio, F. (2015). Wasserstein distances and curves in the Wasserstein spaces. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_5
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