All Questions
Tagged with optimal-transportation oc.optimization-and-control
8 questions with no upvoted or accepted answers
4
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Schrödinger Bridge for other costs
Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23)
$$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t ...
- 101
2
votes
0
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75
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existence of minimizer to the dual problem of a martingale optimal transport type problem
Let $\nu$ be a given probability measure on $\mathbb R^2$ and consider function of the following form:
$$L(f)(x_1,x_2)=\sup_{y\,=\,(y_1,y_2)\,\in\, \operatorname{Graph} (f)} \{ x_1 y_2 + x_2 y_1 - ...
- 325
1
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0
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169
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Optimal transport-like problem where the objective depends on conditional probability distribution
$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data.
Consider two sets $\...
1
vote
1
answer
524
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Optimal transport between two matrices
I'm investigating the use of optimal transport to define a distance between non-negative matrices that satisfy the condition $\mathbf e^\top\mathbf M\mathbf e = 1$ (i.e., the sum of all elements in ...
- 243
1
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0
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85
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Producing a minimiser for the Kantorovich problem from a minimiser of the Beckmann flow problem
Notation: We denote by $\mathcal M$ the set of vector valued measures on $\mathbb R^d$ whose divergence is a scalar measure (in the weak sense).
Definitions: Consider the Beckmann flow minimisation ...
- 6,313
1
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0
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96
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Relationship between Wasserstein projections and metric projections in a linear space
Let $(X,d)$ be a metric space, $x\in X$, and $Y\subset X$ is a closed set. Assume that $X$ is also a real vector space, so that we can form linear combinations over $\mathbb{R}$, but I do not assume ...
- 13
1
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0
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543
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Variational derivative of Wasserstein distance using Benaumou-Brenier formulation
I learned from the gradient flow theory in Wasserstein space that an equation of gradient flow type
$$\partial_t \rho + \nabla \cdot (\rho \nabla \frac{\delta F}{\delta \rho})=0,$$
can be derived as ...
- 429
0
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0
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39
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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 ...