Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \sum_{i=1}^n a_i \delta_{x_i}$ and $\nu\triangleq \sum_{i=1}^n b_i \delta_{y_i}$. Are there known, (not too lax) upper-bounds for $$ W_p(\mu,\nu) \leq M\left(x_1,\dots,x_n,y_1,\dots,y_n,a,b\right) $$ for some continuous function $M$; known in closed-form such that $$ M(x_1,\dots,x_n,x_1,\dots,x_n,a,b)=0. $$

Where $W_p$ is the Wasserstein-1 distance, for some $1\leq p<\infty$?

  • $\begingroup$ The Wasserstein distance is an infimum, so should be bounded by any feasible matching. Any feasible assignment based on a greedy approach will make do. Control by weighted total variation (see the book by Villani, 2008) could also help. $\endgroup$ – Gilles Mordant Oct 23 '20 at 13:19
  • $\begingroup$ The worst you can do, is the 'democratic' coupling $\pi := \sum_{i,j} a_ib_i\delta_{x_i}\otimes \delta_{y_j}$, which gives you the upper bound $$M= ( \sum_{i,j} a_ib_j |x_i-y_j|^p )^{\frac{1}{p}}.$$ $\endgroup$ – Francesco Nobili Oct 23 '20 at 13:49
  • $\begingroup$ I refined my question since there are too many "lax" upper-bounds available... $\endgroup$ – BLBA Oct 23 '20 at 17:09

Let $$F(x):=\mu((-\infty,x])=\sum_i a_i\,1(x_i\le x) =\sum_{j=1}^n s_j\,1(x_{n:j}\le x<x_{n:j+1}),$$ where $x_{n:1}<\cdots<x_{n:n}$ are the values $x_1,\dots,x_n$ put in the increasing order (with $x_{n:n+1}:=\infty$), $$s_j:=\sum_{i=1}^j a_{n:i},$$ and $a_{n:1},\dots,a_{n:n}$ are the values $a_1,\dots,a_n$ put in the increasing order of the $x_k$'s, so that, if $x_{n:i}=x_k$ for some $k$, then $a_{n:i}=a_k$. So, $F$ is the cdf of the probability measure $\mu$. Similarly considered is the function $G$ defined as the cdf of the probability measure $\nu$.

Consider then the generalized inverse/quantile function $F^{-1}\colon(0,1)\to\mathbb R$ defined by $$F^{-1}(u):=\inf\{x\in\mathbb R\colon F(x)\ge u) \\ =\max\{x\in\mathbb R\colon F(x)\ge u) \\ =\sum_{j=1}^n x_{n:j}\,1(s_{j-1}<u\le s_j)$$ for $u\in(0,1)$, with the similarly defined and considered $G^{-1}$. Let $U$ be a random variable (r.v.) uniformly distributed on $(0,1)$. Then the distributions of the r.v.'s $X:=F^{-1}(U)$ and $Y:=G^{-1}(U)$ will be $\mu$ and $\nu$, respectively. Finally, let $$M(x_1,\dots,x_n,y_1,\dots,y_n,a,b):=M(\mu,\nu) \\ :=(E|X-Y|^p)^{1/p} =\Big(\int_0^1|F^{-1}(u)-G^{-1}(u)|^p\,du\Big)^{1/p}.$$ Then $$W_p(\mu,\nu)\le M(x_1,\dots,x_n,y_1,\dots,y_n,a,b)$$ and
$$M(x_1,\dots,x_n,x_1,\dots,x_n,a,a)=0,$$ as desired.

Remark: The upper bound $M(\mu,\nu)$ given above is actually the exact value of the Wasserstein distance for $p\ge1$, according to the last sentence of Theorem 2.1 -- thank you alesia for this reference.

  • $\begingroup$ Why did you only state an inequality? Isn't that the exact value of the Wasserstein distance (you missed an exponent p btw)? $\endgroup$ – alesia Oct 23 '20 at 18:33
  • $\begingroup$ @alesia : The upper bound $M(\mu,\nu)$ given in the answer will in general be the exact value of the Wasserstein distance only for $p=1$ -- see e.g. arxiv.org/pdf/1912.04945.pdf $\endgroup$ – Iosif Pinelis Oct 23 '20 at 18:53
  • $\begingroup$ The first equation p 10 in math.cmu.edu/~mthorpe/OTNotes says it's true for all p $\endgroup$ – alesia Oct 23 '20 at 19:19
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    $\begingroup$ For higher dimensions one could use that $W_p(\mu,\nu)\geq \sup\{\int f\,d(\mu-\nu)\mid Lip(f)\leq 1\}$ so any Lipschitz-1 function gives a lower bound. One may construct some $f$ in some greedy fashion (this does not lead to an "explicit" formula, but a straightforward algorithm for a lower bound). $\endgroup$ – Dirk Nov 5 '20 at 11:56
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    $\begingroup$ (The result is in "On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces" by Gelbrich) $\endgroup$ – Dirk Nov 5 '20 at 11:57

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