# Compute optimal transport costs for signed measures Compute the optimal transport (OT) cost between signed measures that have the same total mass. ```r ot_cost_sgn(mu, nu, costm, mode = c("all", "diag")) ``` ## Arguments - `mu`: matrix (row-wise) or list containing $K_1$ vectors of length $N$. - `nu`: matrix (row-wise) or list containing $K_2$ vectors of length $N$ or `NULL`. - `costm`: cost matrix $c \in \mathbb{R}^{N \times N}$. - `mode`: controls which of the pairwise OT costs are computed. ## Returns The OT cost between the vectors in `mu` and `nu`. For `mode = "all"` the whole matrix of size $K_1 \times K_2$ is returned. If `mu` or `nu` is a vector, then this matrix is also returned as a vector. `nu = NULL` means that `nu = mu` and only the lower triangular part is actually computed and then reflected. If `mode = "diag"`, then only the diagonal is returned (requiring $K_1 = K_2$). ## Details The extended OT functional for vectors $\mu,\,\nu \in \mathbb{R}^N$ with $\sum_{i=1}^N \mu_i = \sum_{i=1}^N \nu_i$ is defined as $$ \mathrm{OT}^{\pm}_c(\mu, \nu) := \mathrm{OT}_c(\mu^+ + \nu^-, \, \nu^+ + \mu^-)\,, $$ where $\mu^+ = \max(0, \mu)$ and $\mu^- = -\min(0, \mu)$ denote the positive and negative part of $\mu$, and $\mathrm{OT}_c$ is the standard OT functional. To compute the standard OT, the function `transport::transport` is used. The values may be computed in parallel via `future::plan`. ## Examples ```r # enable parallel computation if (requireNamespace("future")) { future::plan(future::multisession) } # generate random signed measures with total mass 0 (row-wise) rsum0 <- \(K, N) { x <- runif(K * N) |> matrix(K, N) x <- sweep(x, 1, rowSums(x) / N, "-") x[, 1] <- x[, 1] - rowSums(x) x } K1 <- 3 K2 <- 2 N <- 4 costm <- cost_matrix_lp(1:N) set.seed(123) mu <- rsum0(K1, N) nu <- rsum0(K2, N) print(ot_cost_sgn(mu[2, ], nu[2, ], costm)) # mode = "diag" requires K1 = K2 print(ot_cost_sgn(mu[1:2, ], nu, costm, mode = "diag")) print(ot_cost_sgn(mu, nu, costm)) # only works properly if costm is semi-metric print(ot_cost_sgn(mu, NULL, costm)) # but it requires less computations than print(ot_cost_sgn(mu, mu, costm)) ``` ## See Also `transport::transport`

ot_cost_sgn function