Brenier optimal transport map in one dimension

Formula: T(x_(i)) = y_(i): the monotone rearrangement matching order statistic to order statistic

Usage

Brenier1d(x, y, p = 2)

Arguments

x

See Usage.

y

See Usage.

p

See Usage.

Value

List with map (image of each input in its original order), cost, order_x, order_y, n.

References

Brenier (1991), Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics 44:375-417. The one-dimensional monotone-rearrangement solution is the standard published result.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Brenier1d(V, V)
#> $map
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $cost
#> [1] 0
#> 
#> $order_x
#> [1] 0 1 2 3 4 5 6 7
#> 
#> $order_y
#> [1] 0 1 2 3 4 5 6 7
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Brenier map in one dimension (monotone rearrangement)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"