# Questions tagged [optimal-transportation]

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### Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment

For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
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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 ...
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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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### Generalised Lebesgue transform continuous wrt. strict topology?

Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$]. For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, consider the ...
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### Continuity of generalised Legendre transform

Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$]. For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, we consider ...
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### Weight transfer proof of Turán’s theorem

Turán’s theorem, which states that a $K_{p+1}$-free graph contains at most $(1-1/p)\frac{N^2}{2}$ edges, can be proven in many different ways, as pointed out, for example in M. Aigner, G. M. Ziegler, ...
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### Local Lipschitzness of parameterization of Gaussians in Wasserstein space

Fix a positive integer $n$ and consider the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$. Let $X$ be the cone of $n\times n$ symmetric positive semidefinite matrices with Frobenius norm and ...
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### Max diameter of random partition arising from optimal transport

Let $\mathbf S^2\subset \mathbb R^3$ be the sphere centered at the origin with radius chosen such that $m(\mathbf S^2)=n$, where $m$ denotes the surface area measure. Consider the Wasserstein distance ...
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### Log Sobolev inequality uniform in parameters

Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
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### Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?

Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
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1 vote