Questions tagged [optimal-transportation]
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How to deal with minimizing a flat objective function
Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
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Should coffee machines be deconcentrated?
We model some region by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the people living on $E$, of capacities $\alpha_1,\ldots, \alpha_N>0$. Assume the ...
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Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
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Connection between Wassertein-2 metric and difference in variance
Given two probability densities $\mu\in\mathcal P(\mathbb R^d)$ and $\nu\in\mathcal P(\mathbb R^d)$, we define their Wasserstein-$p$ metric as
$$
W_p^p(\mu, \nu)=\inf_{\gamma\in \Gamma(\mu, \nu)}\int_{...
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Sequential compactness of a sequence of curves of Borel probability measures
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Is this set $\sigma$-compact in the Wasserstein space?
This is a follow-up to this question.
Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:
$$A:=\left\{p\in\mathcal{P}...
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Is the following set compact w.r.t. the Wasserstein distance?
Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:
$$A:=\left\{p\in\mathcal{P}_1(\mathbb R ^d): \int |x|dp\leq K, \...
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Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?
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Optimal transport plan induced by an optimal transport map
I am reading Santambrogio's book on optimal transport, remark 1.19. Let's consider an optimal transport problem between $(X,\mu)$ and $(Y,\nu)$.
(Remark 1.19) ... every time that we know that any ...
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Random solutions to non-Lipschitz ODEs, optimal transport, and general solutions to the continuity equation
I am reading Cedric Villani’s book “Optimal Transport: old and new” and I am stuck on one paragraph (see page 26/27 in this book). He speaks about random solutions to an ODE and I simply cannot figure ...
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Does this maximisation problem admit a finite upper bound?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
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If $\pi$ is a coupling between $f_*\mu_X, g_*\mu_Y$ and $\pi = (f(x),g(y))_* \sigma$, then $\sigma$ is a coupling between $\mu_X,\mu_Y$
I am trying to read this paper: "The Gromov-Wasserstein distance between networks and stable network invariants" https://arxiv.org/abs/1808.04337. In this paper, they have the following ...
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Does complete and separable Wasserstein space imply a complete base space?
Also asked on math.SE.
Let $(Z,d)$ be a metric space, and for $p\geq 1$, consider a metric space $(W_p,d_{W_p})$ defined by
The Wasserstein Space $\begin{align}W_p = \{\mu|\mu\textrm{ is a Borel ...
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Wasserstein bounds of interpolation measures
Assume we are given two densities, $p_0$ and $p_1$ on $\mathbb{R}^d$, and define (up to the normalization constant) the interpolation $p_t \propto p_0^{1-t} p_1^t$, which interpolates between $p_0$ ...
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Any useful bases for the topology induced by the $t$-Wasserstein distance?
I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
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