Wasserstein distance on the line, by sorting

In one dimension the optimal coupling is the monotone one, so the whole linear program collapses to a pair of sorts: the k-th smallest goes to the k-th smallest. This is not an approximation, and it is the reason every sliced method exists.

Usage

Otws2(x, y, p = 2)

Arguments

x, y: Two samples of equal length.
p: Exponent, positive.

Value

List with Wp, Wp_p, n, p.

Details

Formula: W_p^p = (1/n) sum_i |x_(i) - y_(i)|^p for two equal-weight samples – Bobkov and Ledoux (2019) Section 2; Peyre and Cuturi (2019) Remark 2.30.

References

Bobkov, S. and Ledoux, M. (2019). Memoirs of the American Mathematical Society 261(1259). doi:10.1090/memo/1259 .

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Otws2(V, V)
#> $Wp
#> [1] 0
#> 
#> $Wp_p
#> [1] 0
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 2
#> 
#> $method
#> [1] "Univariate Wasserstein distance"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"