All Questions
Tagged with optimal-transportation convex-optimization
11 questions
1
vote
1
answer
60
views
Optimal transport for sum of two costs
Let $X$ be a finite set and $\sigma_0$, $\sigma_1$ two fixed measures on $X$ with $\sigma_0(X)=\sigma_1(X)$. A transportation plan is a measure $\mu$ on $X\times X$ whose projections on the first and ...
- 1,074
1
vote
1
answer
108
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
0
votes
0
answers
54
views
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 ...
- 121
3
votes
0
answers
87
views
Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
- 232
2
votes
2
answers
293
views
Optimal transport: the existence of an optimal pair of $c$-conjugate functions
$\newcommand{\diff}{ \, \mathrm d}$
Let
$X,Y$ be Polish spaces,
$\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$,
$\mathcal P(X)$ the space of Borel probability ...
- 825
1
vote
0
answers
37
views
Sufficient condition for an $n$-tuple to be a convex conjugate
We say $(f_1,f_2,\dotsc,f_N)$ is a convex conjugate if for any $i=1,2,\dotsc,N$ and any $x_i\in\Bbb R^d$, we have:
$$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(...
- 183
1
vote
0
answers
113
views
Statistical analysis of optimization solution involving Brenier potentials?
I'm reading the paper https://arxiv.org/pdf/1905.10812.pdf where strongly convex approximations to Brenier potentials are approximated.
Let $\mathcal{E}$ be a partition of $\mathbb{R}^{d}$ and $ 0\leq ...
- 383
3
votes
1
answer
170
views
Reformulation as optimization on probability distributions
This is a "soft" question, in the sense that I'm looking for historical remarks and general commentary rather than a definite answer.
For compact $X \in R^n$ and $f : R^n \to R$ consider the problem
...
4
votes
1
answer
351
views
Inf of Jensen's inequality
I'm reading a monograph that considers the following problem:
$$\inf_{z(t) \in C^1} \int_0^1 c\bigg(\frac{dz(t)}{dt}\bigg) dt\\ z(0) = x, z(1) = y$$
Here $c$ is a convex function, $z(t)$ are paths ...
- 427
6
votes
2
answers
2k
views
Wasserstein distance and the Kantorovich-Rubinstein duality
The only few references I could find on this topic are either amateur blog posts (http://n.ethz.ch/~gbasso/download/A%20Hitchhikers%20guide%20to%20Wasserstein/A%20Hitchhikers%20guide%20to%...
- 2,246
1
vote
1
answer
129
views
Optimization problem restricted to a smaller field?
Let $c:\mathbb R^2\to\mathbb R$ be a Lipschitz and bounded function (which can be supposed as "nice" as possible). Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ with finite first ...
- 1,835