All Questions
Tagged with optimal-transportation measure-concentration
9 questions
4
votes
0
answers
107
views
Upper bound $\tau_C := \int_{\|x\| \le 1}(vol(C \cap (x + C))/vol(C))dx$ for a convex body $C \subseteq \mathbb R^n$, by reducing to a ball
Let $C$ be a convex body in $\mathbb R^n$, i.e a bounded convex subset of $\mathbb R^n$ which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the ...
- 6,853
4
votes
0
answers
183
views
Regularity of optimal transport of Gaussian mixtures
In one of the problems that I am working on, I came across the topic of smoothness of optimal transport for Gaussian mixtures. In particular, let $P=P_\theta=\sum_{i=1}^k \frac{1}{k}\mathcal{N}(x| \...
- 161
3
votes
1
answer
530
views
Wasserstein-type concentration inequalities for empirical measures on polish spaces
Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space ...
- 5,405
2
votes
1
answer
304
views
An approximation problem w.r.t marginal distribution of coordinates of uniform random vector on high-dimensional unit-sphere
Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define ...
- 6,853
2
votes
1
answer
114
views
Reweighting probability measures by convex potentials, and contraction in transport distance
Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by
$$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$
i.e. each $\mu_y$ is a ...
- 801
2
votes
0
answers
131
views
Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
- 6,853
1
vote
1
answer
685
views
Existence of optimal coupling in optimal transport
Let $P,Q$ be any two distributions over a space $\mathcal{X}$ and let $\mathcal{M}(P,Q)$ be the set of all couplings of $P$ and $Q.$ For a given metric $d$ over $\mathcal{X},$ the optimal transport ...
- 1,269
1
vote
1
answer
524
views
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
vote
0
answers
176
views
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