Questions tagged [wasserstein]
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66
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Is the space of probability measure dense in the space of probability measure with density?
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of all probability measures with finite second moment. Let $\mathcal{P}_2^*(\mathbb{R}^n)$ be the subset of $\mathcal{P}_2(\mathbb{R}^n)$ such that any ...
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Relationship between optimal transport and gaussian kernel
Let's say P and Q be two different dirac delta probability measure, and suppose that $K_\sigma$ is a gaussian kernel. Let D be the wasserstein-2 distance.
It is known that $D(P,Q)=D(K_\sigma *P, K_\...
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Wasserstein distance between a distribution of a random variable and the distribution of its projection onto a subset of its sample space
Consider a random variable $x$ with a distribution $p_x$ supported on whole of $\mathbb{R}^n$ ($n$ being a natural number). Let $S \subset \mathbb{R}^n$. Let $y = {\rm proj}_S(x)$ denote the ...
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Does the reverse triangle inequality holds for Wasserstein-1 distance?
Let $(X,d)$ be a separable metric space with associated Borel $\sigma$-algebra $\mathcal{B}(X)$ and the set of Borel probability measures $\mathcal{P}(X)$. For $\mu,\mu'\in\mathcal{P}(X)$ Wasserstein-...
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Wasserstein distance inequality
Suppose $(\Omega, \mathcal F, \mathbb P)$ is a probability space. Suppose $X, X', Y, Y'$ are random variables.
Denote $W_1$ the Wasserstein-1 distance between $\mathbb P_X$ and $\mathbb P_{X'}$
and $...
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An inequality about the 2-Wasserstein distance
Let $W_2(\mu,\nu)$ denote the $2$-Wasserstein distance between two given probability measures $\mu$ and $\nu$ on $\mathbb R^n$. For a probability measure $\mu$ and $f:\mathbb R^n\to \mathbb R^n$, let $...
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Scaling property of the Wasserstein metric
I would need help with this example.
Let $(S, ||\cdot||)$ denote a normed vector space over $K =\mathbb R$ or $K =\mathbb C$. Let $X$ and $Y$ be $S$-valued random vectors with $E~[~||X||~] < \infty$...
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Advection reaction equation is solved by projection of solution of continuity equation
Suppose an absolutely continuous curve $\mu \colon (0, \infty) \to P_2(\Omega)$, where $P_2$ is the Wasserstein-2-space, fulfils the continuity equation
$$ \label{eq:CE} \tag{CE}
\partial_t \mu_t = \...
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Wasserstein-1 distance, $W(A\cdot B \| C\cdot D)=W(A \| C) + W(B \| D)$
For 4 independent random variables $A, B, C, D$ and Wasserstein-1 distance $W^1$,
$W^1(P_{A,B} \parallel P_{C,D})=W^1(P_A \parallel P_C)+W^1(P_B \parallel P_D)$
Does the above equation generally ...
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Optimal Transport between two Gaussians
Consider the optimal transport map $T$ between $N(\mu_0,\Sigma_0)$ and $N(\mu_1,\Sigma_1)$. I believed that the optimal transport was given by:
$$ T(x) = \mu_1 + \Sigma_1^{1/2} \Sigma_0^{-1/2}(x-\mu_0)...
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Why does it suffice to define the homogeneous projection operator by only testing with continuous instead of measurable functions?
In Lenaic Chizat's "Sparse Optimization on Measures with Overparametrized Gradient Descent" one finds the following definition: for a measure $\mu \in \mathcal P_2(\Omega)$ (the Wasserstein-...
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Can we bound the L1 distance between densities by Wasserstein distance of measures
Let $\mu_1$ and $\mu_2$ be two probability measures over a closed interval $[a, b]$, with respective density functions $\phi_1$ and $\phi_2$. Is there a way to bound the $L^1$ distance of the ...
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Absolutely continuous curves in Wasserstein distance and measurability.
Let $(X, d, \mu)$ be a metric measure space. Let $P^1(X)$ denote the space of probability measures on $(X,d)$, which have finite first moments, that is:
\begin{equation}
\nu \in P^1(X) \implies \int d(...
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Weighted median of distribution functions
I am working on the following barycenter problem: Suppose we are given $N>1$ probability measures on $\mathbb{R}$ with cumulative distribution functions $F_1,\dots,F_N$ and weights $a_1, \dots, a_N ...
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Is the median of CDFs again a CDF?
I am working on the following barycenter problem: Suppose we are given $N$ probability measures on $\mathbb{R}$ with cumulative distribution functions $F_1,\dots,F_N$ and we are interested in the ...
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