I have two sets of $n$ $d$-dimensional vectors $X$ and $Y$. So I can compute the Earth Mover's Distance (EMD), or Wasserstein metric, between them:
$\delta_1 = \operatorname{EMD}(X, Y)$
Now I have two groups of three sets of $n$ vectors $X_1, X_2, X_3$ and $Y_1, Y_2, Y _3$.
I want to compute $\delta_2 = \operatorname{EMD}(\cdot)$ between all $X_i$ and all $Y_i$ as defined above. How do I note that these sets are combined? If I would write $X_1 ∪ X_2 ∪ X_3$, I would end up with less than $3n$ elements if two of the three sets contain the same element.
I actually need to do a weighted EMD computation, which means that all elements of $X_1, Y_1$ are assigned weight $2$, while for all others, the weight is $1$. I could define a weight vector $w = (2 ··· 2, 1 ··· 1, 1 ··· 1)$ that is symmetrically used for both inputs of $\operatorname{EMD}_w(\cdot)$. However for this to work, the sets need to stay in order.
Is there any concise notation for this?
If i can understand your question, the best would be to define everything in terms of sequences : $x_1=(x_1^k)_{1\leq k \leq n}$, $x_2=(x_2^k)_{1\leq k \leq n}$, $x_3=(x_3^k)_{1\leq k \leq n}$, $y_1=(y_1^k)_{1\leq k \leq n}$, $y_2=(y_2^k)_{1\leq k \leq n}$, $y_3=(y_3^k)_{1\leq k \leq n}$, define the concatenation operation : $$(x_1,\ldots,x_p) \circ (y_1,\ldots,y_q) = (x_1,\ldots,x_p,y_1,\ldots,y_q)$$ and the quantity you're looking for should be $$EMD(x_1\circ x_1 \circ x_2 \circ x_3, y_1\circ y_1 \circ y_2 \circ y_3).$$