Questions tagged [optimal-transportation]
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94 questions with no upvoted or accepted answers
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a generalization of Monge-Kantorovich Problem
I am thinking about the martingale version of Monge-Kantorovich Problem.
Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on $\...
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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
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1
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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
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433
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Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)
Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on ...
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96
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Representation formula for the continuity equation on a separable Hilbert space
The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (...
- 111
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170
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$L^2$ metric on $\textrm{Diff}(M)$ and geodesics
The paper Geometry of diffeomorphism groups, complete integrability and optimal transport mentions the following:
The group $\textrm{Diff}(M)$ carries a natural $L^2$-metric
$\displaystyle \langle\...
- 305
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70
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$c$-cyclical monotonicity: does this proof hold if $f \equiv +\infty$ or $\int c \mathrm d \gamma = +\infty$?
I'm reading the proof of Theorem 1.38. from section 1.6.2 $c$-cyclical monotonicity and duality of Santambrogio's Optimal transport for applied mathematicians.
My understanding: It seems for the ...
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82
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Conditions for existence of an optimal transport map between finitely supported probability measures
Let $E$ be a polish and let $P$ and $Q$ be finitely supported probability measures on $X$. What conditions are required to ensure that: for every $\delta>0$ there exists a $\delta$-optimal ...
- 11
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1
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125
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Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
- 63
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Transport-type duality for preduals of $C^{k,1}$-functions
Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
- 5,405
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193
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What is the role of of continuity in this proof of Kantorovich duality?
I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation.
Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability measures ...
- 825
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256
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Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
- 5,405
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71
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Hyperplanes which equalize the Radon transforms of two distributions
Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
- 6,853
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37
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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
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130
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Orthogonality in Wasserstein tangent space for discrete measures with equal mass
Let say I have $N$ discrete probability measures $(\mu_1,...,\mu_N)$ where each of them has $n$ points in $\mathbb{R}^2$ of equal mass.
Let $P(\mathcal{X})$ be the space of these probability measures ...
- 19
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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
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113
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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
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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
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56
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Lax CD(K, $\infty)$ space in the sense of Sturm
In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\...
- 660
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Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
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113
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Metrics on the space of distributions in terms of p.d.fs
If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm ...
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Transport of measure
Let's disintegrate $\mu$ and $\nu$, two probabilities on $\mathbb{R}^{d}$ , according to
$$
\pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d})
$$
We get a family of measures and each measure $\mu_{k,d}^{+...
- 179
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543
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Variational derivative of Wasserstein distance using Benaumou-Brenier formulation
I learned from the gradient flow theory in Wasserstein space that an equation of gradient flow type
$$\partial_t \rho + \nabla \cdot (\rho \nabla \frac{\delta F}{\delta \rho})=0,$$
can be derived as ...
- 429
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0
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81
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Empirical estimation of $\inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega)$, given i.i.d samples from $\mu$ and $\nu$
Let $\mathcal X$ be a Polish space and $\Omega \subseteq \mathcal X^2$ be open. Let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by
$$
c_\Omega(\mu,\...
- 6,853
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176
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Rate of convergence of empirical distribution with respect to Wasserstein distance induced by binary cost function
Let $\mathcal X=(\mathcal X, d)$ be a Polish space (i.e complete metric space), and let $\Omega$ be a non-empty subset. Consider the binary cost function $c_\Omega$ on $\mathcal X^2$ defined by $c_\...
- 6,853
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60
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Euler-Lagrange Equation for Kantorovich Dual Problem
Given two probability measures $\mu$ and $\nu$, the Kantorovich Dual problem for quadratic cost is to
$$
\text{minimize} \quad \int \phi(x)d\mu + \int \psi(y)d\nu
$$
over pairs $(\phi,\psi)\in L^1(d\...
- 325
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97
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Dependency of the Wasserstein metric on its parameters
Let the population on some region $\Omega\subset\mathbb R^d$ be modeled by a density function $\rho:\Omega\to (0,+\infty)$. Provided $n\ge 1$ food trucks labeled by their capacity $p_1,\ldots, p_n\in (...
user128095
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Negative Definiteness of Hopf-Lax-Oleinik Semigroup
Denote by $H_{t}$ the Hopf-Lax semigroup, i.e.\begin{equation}
H_{t}f(x)=\inf_{y\in\mathbb{R}}\left\lbrace f(y)+\frac{(x-y)^{2}}{2t}\right\rbrace.\end{equation} Is $H_{t}$ negative definite on bounded,...
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$\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E
Given two random variables X,Y with measures P,Q. Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then $\mathbb{P}(d(X,Y)>\alpha)<\beta$.
Only hints please.
...
- 395
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Constructing a family of domains for application of method of continuity in optimal transportation
Anyway can help me about this paper?
http://arxiv.org/pdf/math/0601086v4.pdf
I want to ask page 20, The author want to construct a family of subdomain for using method of continuity. But I can't ...
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52
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Path-homotopy in Wasserstein space
Consider two vector fields $b_0,b_1\in C^2([0,1]\times\mathbb{R}^d;\mathbb{R}^d)$ and the solutions $\rho_0,\rho_1\in AC([0,1];\mathcal{P}_2(\mathbb{R}^d))$ to the associated Fokker-Planck equations
$$...
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54
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How to deal with minimizing a flat objective function
Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
- 121
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Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
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55
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Any useful bases for the topology induced by the $t$-Wasserstein distance?
I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
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114
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Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
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114
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Wasserstein compactness of sublevel sets of relative entropy
Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\...
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"More" cyclical monotonicity
Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\...
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Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
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291
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Convexity of a set of probability densities
Consider the space of probability densities $(P(\mathbb{R}^d), W_2)$ (probability measures on $\mathbb{R}^d$ with 2-Wasserstein distance).
How can we determine if a subset $Q$ is convex?
I know that a ...
- 153
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How does one define weak convergence of probability measures in $L^{\infty}(\Omega)$?
I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $\gamma_{\epsilon_k}|\_G_{\delta}\times \Omega\to \gamma|\_G_{\delta}\times \Omega$ as $\epsilon_k\to 0$ ...
- 547
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72
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Optimal transport mapping between sets with a common subset
What would the optimal transport mapping between two sets with a common subset be? The problem I'm thinking about is the following:
I'm in $\mathbb{C}^n$ and I have two distributions $\mu$ and $\nu$ ...
- 383
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75
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Optimizer of a semi-discrete optimal transport problem
Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-...
user128095
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103
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Expectation of maximal Wasserstein distance between empirical distribution and a pdf
Let $P$ be a continuous probability distribution on $R^d$, $X$ the random variable $\sim P$, and $
\hat{X}$ be n i.i.d samples drawn according to $P$. We have another variable $\mu \in S^{d-1}$.
Do ...
- 109
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184
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
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