dg.differential geometry – Orthogonality in Wasserstein tangent space for discrete measures with equal mass

Let say I have $N$ discrete probability measures $(mu_1,…,mu_N)$ where each of them has $n$ points in $mathbb{R}^2$ of equal mass.

Let $P(mathcal{X})$ be the space of these probability measures on a Hilbert space $mathcal{X}$ (which is $mathbb{R}^2$ in our case if I’m not mistaken).

We can define the Wasserstein barycenter:

$overline{mu} = argmin_{nu in P(mathcal{X})} sum_i^N alpha_i W_2^2(mu_i, nu)$.

Under the 2-Wasserstein metric $W_2$ and with $alpha_i = frac{1}{N},_{i=1..N}$ (in our case).

I’m interested to grap the intuition with the tangent space at $overline{mu}$. Especially in the notion of orthogonality in this tangent space.

Let say I have a vector (or velocity) field $v_1 in L^2(overline{mu}, mathcal{X})$ (the tangent space at $overline{mu}$), containing $n$ vectors, that move all the $n$ masses of $overline{mu}$ in a direction $v_1^i,_{i=1..n}$ respectively.

What would it mean to have a velocity field $v_2 in L^2(overline{mu}, mathcal{X})$ orthogonal to $v_1$ in the $L^2(overline{mu}, mathcal{X})$ sense? It is just that we need to have $v_2^i perp v_1^i,_{i=1..n}$ in the $mathbb{R}^2$ sense?

I’m interested in any reference that could help (especially because I could need to cite something) and any explications.

Thank you very much.

pr.probability – Wasserstein Continuity of induced map

Setup:
Let $X_1,X_2$, and $X_3$ be polish metric spaces and let $f:X_1times X_2rightarrow X_3$ be continuous. Denote the Borel $sigma$-algebra on $X_i$ by $mathcal{B}(X_i)$, for $i=1,2,3$. Let $(mathcal{X},Sigma)$ be a measurable space and let $g:mathcal{X}rightarrow X$ be $(Sigma,mathcal{B}(X))$ be measurable.
Clearly the map $(xi,x)in mathcal{X}times X_2rightarrow X_3$ is $Sigma$-measurable in the first component and continuous in the second.

Question: If $mathbb{P}$ is a probability measure on $(mathcal{X},Sigma)$; thus $g$ is an $X_2$-valued random element and the collection ${G_xtriangleq g(cdot,x)}_{xin X_2}$ is a family of $X_3$-valued random elements. Under these conditions, is the map:
$$
xmapsto mathbb{P}(G_xin cdot),
$$

continuous wrt the Wasserstein metric on $mathcal{P}(X_3)$?

fa.functional analysis – Wasserstein Space subspace of a Banach Space

Let $X$ be a separable and compact (and therefore complete) metric space. Consider the 1-Wasserstein $P_1(X)$ space lying over $X$ (in the sense that $xmapsto delta_x$ is an isometric embedding). Can $P_1(X)$ be viewed as a convex subset of some separable Banach space $E$? That is, does there exist a separable Banach space $(E,|cdot|_E)$ and an isometry (or at-least a bi-Lipschitz map $phi$)
$$
psi:P_1(X)rightarrow E
$$

such that $psi(P_1)$ is a convex subset of $E$?

I wanted to do something like this with the Arens-Eells space, but the norm is a bit different there…

pr.probability – Comparison of Information and Wasserstein Topologies

There are many possible metrics one can place on the space of Gaussian probability measures on $mathbb{R}^n$, with strictly positive definite co-variance matrices. Let’s denote this space by $X$.

I’m particularly interested in the information geometric one (using the Fisher-Rao-Riemann metric) and the one induced by restricting the Wasserstein $2$ metric from $mathcal{P}_2(mathbb{R}^n)$ to the subspace $X$. But how doe these compare? Most specifically, is the Wasserstein $2$-distance dominated by the Fisher-Rao metric’s induced distance function?

pr.probability – Show that the adjoint of this Markov semigroup eventually preserves a Wasserstein space

Let $E$ be a separable $mathbb R$-Banach space, $mathcal M_1$ denote the set of probability measures on $(E,mathcal B(E))$, $(kappa_t)_{tge0}$ be a Markov semigroup on $(E,mathcal B(E))$ and $rho$ be a metric on $E$ with $$(mukappa_totimesdelta_0)rhole cint v^{lambda(t)}:{rm d}mutag1$$ for all $tge0$ and $muinmathcal M_1$ for some $cge0$, some continuous $v:Eto(1,infty)$ and some nonincreasing $lambda:(0,infty)to(0,1)$ with $lambda(t)xrightarrow{ttoinfty}0$.

Let $$mathcal S^1:={muinmathcal M_1:exists yin E:(muotimesdelta_y)rho<infty}$$ denote the Wasserstein space associated to $rho$. Are we able to conclude that, if $(kappa_t)_{tge0}$ has an invariant measure, it must belong to $mathcal S^1$?

My idea is to show that there is a $t_0ge0$ such that the adjoint$^1$ $kappa_t^ast$ maps $mathcal M_1$ to $mathcal S^1$ for all $tge t_0$.

I’m only able to prove something weaker: Let $muinmathcal M_1$. By the monotone convergence theorem, $$(mukappa_totimesdelta_0)rhoxrightarrow{ttoinfty}ctag2.$$ So, there is a $t_0ge0$ such that $(mukappa_totimesdelta_0)rho<infty$, but since this $t_0$ clearly depends on $mu$, I’m not sure if this is sufficient for the desired conclusion.


$^1$ $kappa_t^astmu:=mukappa_t$.

Probability – rate of convergence of the empirical distribution with respect to the Wasserstein distance induced by the binary cost function

To let $ mathcal X $ be a Polish space (ie complete and measurable), and let $ Omega $ be a non-empty subset. Consider the binary cost function $ C_ Omega $ on $ mathcal X ^ 2 $ defined by $ C_ Omega (x, x & # 39;) = begin {cases} 1, & mbox {if} (x, x & # 39;) in Omega, \ 0, & mbox {else, } end {cases} $
and the induced lattice spacing over probability distributions $ mathcal X $, defined by
$$
C_ Omega (Q_1, Q_2): = inf _ { gamma} mathbb E _ {(x, x & # 39;) sim gamma} (C_ Omega (x, x & # 39;)) = inf _ { gamma} gamma ( omega),
$$

where the infimum takes over everything clutches from $ Q_1 $ and $ Q_2 $,

Finally leave $ P $ Be a probability distribution on this space and leave $ X_1, ldots, X_n sim P $ an egg. d. Be a sample of the size $ n $, and let $ has {P} _n: = (1 / n) sum_ {i = 1} ^ n delta_ {x_i} $ let be the induced empirical distribution.

What are good end limits for random variables? $ c_N (P, has {P} _n) $ ? That is, z $ t> 0 $ What is a good upper limit for the probability $ P (C_N (P, has {P} _n) t t) $ ?