## 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.

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## 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 $$mathcal{P}(X_3)$$?

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## 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…

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## 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?

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## 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 $$mathcal S^1:={muinmathcal M_1:exists yin E:(muotimesdelta_y)rho denote the Wasserstein space associated to $$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 $$(mukappa_totimesdelta_0)rho, but since this $$t_0$$ clearly depends on $$mu$$, I’m not sure if this is sufficient for the desired conclusion.

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

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## 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 $$P (C_N (P, has {P} _n) t t)$$ ?

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