Questions tagged [wasserstein]
The wasserstein tag has no usage guidance.
65
questions
4
votes
0
answers
52
views
Concentration inequalities for random measures
For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality:
$$\mathbb{P}\left(\left|\mu -\frac1n\...
- 1,261
1
vote
0
answers
296
views
Wasserstein distance of convolution of measures
Let $\mu_1, \mu_2,\nu_1,\nu_2$ be measures on $\mathbb{R}^d$. It is well known that the $p$-Wasserstein distance satisfies
$$\mathcal{W}_p(\nu_1 *\mu_1, \nu_1 *\mu_2) = \mathcal{W}_p(\mu_1,\mu_2),$$
...
- 1,261
0
votes
0
answers
271
views
Derivative of the Wasserstein Metric between two Gaussians
I am trying to take the derivative of the squared Wasserstein metric between two Gaussian probability densities, which is given by $W_2^2(q_0, q_1) = \| \boldsymbol{\mu}_0 - \boldsymbol{\mu}_1 \|_2^2 +...
- 41
2
votes
1
answer
58
views
Prove $\frac{1}{n}∑_{k=1}^n \Big(x_k^2 + 2\sqrt{3}(1 - \tfrac{2k-1}{n}) x_k + 1 \Big)>0$ given $x_1<…<x_n$
I know for a fact that
$$\frac{1}{n}∑_{k=1}^n \Big(x_k^2 + 2\sqrt{3}(1 - \tfrac{2k-1}{n}) x_k + 1 \Big)>0 \qquad\text{if $x_1<x_2<…<x_n$}$$
should hold because I derived this sum as the ...
- 11.8k
0
votes
0
answers
42
views
Measuring the similarity between distance vectors
I am trying to measure the correlation between a probability distribution and a scalar value. For instance, I have the following:
Vector of values
Corresponding Scalar
Vec 1
Scalar 1
Vec 2
Scalar 2
...
- 1
1
vote
1
answer
176
views
What is the p-th moment finite in the definition of Wasserstein space?
I am confused about the following notation:
For a simple case, let $X=R^d$ or $X=R$. What dose
$$
\int_X \|x\|^pd\mu(x)
$$
mean for a Borel probability measure $\mu$?
For $X=R$, then $x\in R$ is a 1-...
- 1,520
0
votes
0
answers
22
views
Is there any relationship between the following two expectations?
Is there any relationship between the following two expectations?
$\mathbb{E}_{\mathbb{Q}}[\|\boldsymbol{\tilde{\xi}} - \boldsymbol{\tilde{\xi}}^{\prime}\|]$, and
$\mathbb{E}_{\mathbb{Q}}[\|\mathbf{A}...
- 41
1
vote
0
answers
54
views
Derivative of Kantorovic Potential wrt to Measure
The Kantorovich Dual of the 1-Wasserstein distance $W_1(p,q)$ between two densities $p(x), q(x)$ is given by
$$W_1(p,q) = \sup_{|f|_L\leq 1} \int f(x)(p(x)-q(x))dx$$
with $|f|_L$ denoting the ...
0
votes
1
answer
197
views
Question on Kantorovich-Rubinstein Duality proof
I am currently working on understanding the Kantorovich-Rubinstein duality and Wassertein loss.
The following part of these class notes:
Collecting the terms algebraically we can rewrite the ...
2
votes
1
answer
362
views
Wasserstein metric vs Holder continuity
It is well known that if $f$ is a Lipschitz continuous function, i.e.
$$\forall x,y\in \Omega\qquad |f(x)-f(y)|\le L\|x-y\|$$
then, for any two probability distributions $\mu, \nu$
$$\int_\Omega f(x)(...
- 1,199
0
votes
0
answers
127
views
Calculating the limit of a Wasserstein distance of two SDE's
I am trying to prove that:
$\lim_{t \to \infty} W_2(\mu_t, \nu_t) = 0 $ where we have that $\mu_t = Law(X_t)$ and $\nu_t = Law(Z_t)$ with
$$dX_t = -h(X_t)dt + \sqrt(\frac{2}{\beta})dB_t$$
$$dZ_t = -h(...
- 1
1
vote
0
answers
105
views
2-Wasserstein barycenter of uniform distribution on ellipsoid
Let $A$ be a positive-definite symmetric matrix. Consider the ellipsoid $E = \{ x \in \mathbb{R}^n \colon <x A^{-1} x> \leq 1 \}$.
Now consider uniform distribution $\mu_1, \ldots, \mu_n$ on $...
- 87
0
votes
1
answer
205
views
What is a clever or efficient way to compute this variant of the Wasserstein distance between persistence diagrams?
A two-dimensional persistence diagram in $[0,1]$ say is just a multiset of points of $\mathbb R^2$. Given two diagrams $P=\{p_1=(a_1,b_1),\ldots, p_n= (a_n,b_n)\}$ and $Q=\{q_1=(c_1,d_1),\ldots, q_n= (...
- 10.4k
1
vote
1
answer
100
views
Extension of Kantorovich-Rubinstein inequality.
Let $(\mathcal{X}, \Sigma)$ be a Polish metric space, endowed with the Borel $\sigma$-algebra. Let $\mathscr P$ be the space of probability measures on $\mathcal X$ and $\mathscr P^1$ be defined as
$$\...
- 2,970
4
votes
1
answer
136
views
Maximiser of $W_1(\mu, \nu)$ can be changed outside of $\text{conv}(\text{supp}(\mu) \cap \text{supp}(\nu))$ (under additional assumptions)
Let $(X, \| \cdot \|)$ be a reflexive Banach space and $\mathbb{P}_n$, $\mathbb{P}_r$ be measures on $X$.
Let the support of $\mathbb{P}_r$, $M := \text{supp}(\mathbb{P}_r)$ be a weakly compact set ...
- 4,878