All Questions
Tagged with optimal-transportation riemannian-geometry
9 questions
3
votes
0
answers
49
views
Transport map to lower dimension?
Let $S^{d-1}$ be the sphere in $\mathbb{R}^d$.
Given a $C^\infty$ function $f \colon S^{d-1} \to \mathbb{R}$, define $g \colon S^{d-1} \to S^{d-1}$ as $g(x) = \exp_x(\nabla f(x))$, where $\nabla f(x)$ ...
- 171
3
votes
1
answer
162
views
Gradient flows: evolution of geodesics
I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic
connecting the ...
- 33
1
vote
1
answer
262
views
What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?
I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9):
Here, $M$ is a compact Riemannian manifold, $\...
- 305
4
votes
1
answer
209
views
Riemannian submanifolds of $2$-Wasserstein space
In the article "Wasserstein Geometry Of Gaussian Measures" by Asuka Takatsu the author shows how the space of d-dimensional Gaussian probability measures with non-singular covariance ...
- 91
1
vote
1
answer
192
views
Are convex functions on manifolds the same as $c$-convex functions, where $c(x,y)=d(x,y)^2/2$?
I am reading the following book on optimal transport. While reading I came across the following definition of $c-$convexity.
Given $X$ and $Y$ metric spaces, $c: X \times Y \rightarrow \mathbb{R}$, ...
- 547
3
votes
0
answers
82
views
Semiconcavity estimate for the squared distance on a compact Riemannian manifold
I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...
- 31
3
votes
2
answers
336
views
Invertibility of neural network as operator on Wasserstein space
Question statement: Consider the space of probability measures with finite second moments $P_2(\mathbb{R}^d)$, which is equipped with the Wasserstein-2 distance $W_2$, and the square integrable ...
- 1,127
9
votes
2
answers
778
views
Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity
In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
- 5,380
7
votes
0
answers
433
views
(geodesic) smoothness of f-divergence with respect to the Wasserstein metric
We consider the f-divergence, which takes the form
$$
D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ.
$$
For example, when $f(t) = t \log t$, we obtain the KL-divergence.
My question is ...
- 1,127