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5 votes
0 answers
158 views

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 $x_1,\ldots,x_N\in \mathbb{R}^2$,...
Matt Rosenzweig's user avatar
4 votes
0 answers
116 views

Improving log-Sobolev inequalities via quadratic regularisation

Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$. For suitable functions $g \geqslant 0$, define $$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
πr8's user avatar
  • 801
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 ...
πr8's user avatar
  • 801
0 votes
2 answers
532 views

Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$

Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$. Question. What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
dohmatob's user avatar
  • 6,853