What is Optimal Transport?

Imagine you have two piles of sand with different shapes, and you want to reshape one pile to match the other. Optimal transport (OT) finds the cheapest way to move the sand — that is, the plan that minimizes the total cost of moving mass from one distribution to another.

In the simplest case, the two distributions are one-dimensional histograms. The “transport plan” is a matrix that tells you how much mass to move from each bin of the source to each bin of the target.

# Two simple 1D distributions
source_dist <- c(0.4, 0.1, 0.4, 0.1)
target_dist <- c(0.1, 0.3, 0.1, 0.3, 0.2)

oldpar <- par(mfrow = c(1, 2), mar = c(4, 4, 3, 1))
barplot(source_dist, col = "steelblue", main = "Source distribution",
        names.arg = seq_along(source_dist), ylim = c(0, 0.5),
        xlab = "Bin", ylab = "Mass")
barplot(target_dist, col = "tomato", main = "Target distribution",
        names.arg = seq_along(target_dist), ylim = c(0, 0.5),
        xlab = "Bin", ylab = "Mass")

par(mfrow = oldpar)

OT finds a transport plan (a matrix) that optimally maps the source to the target. Each cell of the matrix represents how much mass moves from a source bin to a target bin.

What is a Tensor?

A tensor is simply a generalization of familiar data structures:

• Order 1 (vector): a list of numbers, e.g., temperature readings over time
• Order 2 (matrix): a table of numbers, e.g., pixels in a grayscale image
• Order 3 and higher: a “cube” or higher-dimensional array, e.g., a color image (height x width x RGB channels)

oldpar <- par(mfrow = c(1, 3), mar = c(2, 2, 3, 1))

# Vector (order 1)
barplot(c(3, 1, 4, 1, 5), col = "steelblue", main = "Order 1: Vector")

# Matrix (order 2)
mat <- matrix(c(1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6), nrow = 3)
image(mat, col = gray((0:255) / 255), axes = FALSE, main = "Order 2: Matrix")

# 3D tensor (show one slice)
arr <- array(0, dim = c(3, 4, 2))
arr[,,1] <- matrix(c(1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6), nrow = 3)
arr[,,2] <- matrix(c(6, 5, 4, 3, 5, 4, 3, 2, 4, 3, 2, 1), nrow = 3)
image(arr[,,1], col = gray((0:255) / 255), axes = FALSE,
      main = "Order 3: Tensor\n(slice 1)")

par(mfrow = oldpar)

The Problem OTT Solves

Standard OT works well for vectors and matrices, but what if your data is a higher-order tensor?

Optimal Tensor Transport (OTT) (Kerdoncuff 2022) extends OT to tensors of any order. Given two tensors \(X\) and \(Y\) of the same order, OTT finds transport plans — one or more matrices that describe how to map each dimension of \(X\) to the corresponding dimension of \(Y\).

The Key Concept: the f Parameter

The f parameter is the core idea that makes OTT flexible. It is a vector that assigns each dimension to a transport plan group. This controls how dimensions share transport plans.

For example, with a 3D tensor (e.g., subjects x genes x time):

• f = c(1, 2, 3) learns separate transport plans for subjects, genes, and time
• f = c(1, 1, 2) forces subjects and genes to share a plan, while time has its own

Quick Start Example

Here we walk through a minimal example step by step.

library("otTensor")
library("rTensor")

# Source: a 4 x 5 matrix
arrX <- matrix(0, nrow = 4, ncol = 5)
for (i in 1:4) {
    for (j in 1:5) {
        arrX[i, j] <- i + j
    }
}

# Target: a 6 x 7 matrix (different size is OK)
arrY <- matrix(0, nrow = 6, ncol = 7)
for (i in 1:6) {
    for (j in 1:7) {
        arrY[i, j] <- i + j
    }
}

# Convert to Tensor objects
X <- as.tensor(arrX)
Y <- as.tensor(arrY)

f <- c(1, 2)

result <- OTT(X = X, Y = Y, f = f,
              num.sample = 500, num.iter = 100)

# Transport plan dimensions
cat("Transport plan 1:", dim(result$Ts[[1]]), "\n")

## Transport plan 1: 4 6

cat("Transport plan 2:", dim(result$Ts[[2]]), "\n")

## Transport plan 2: 5 7

.show_matrix <- function(mat, main = "") {
    mat_rev <- t(apply(mat, 2, rev))
    image(mat_rev, col = gray((0:255) / 255),
          xaxt = "n", yaxt = "n",
          xlab = "", ylab = "", axes = FALSE, main = main)
}

oldpar <- par(mfrow = c(2, 2), mar = c(2, 2, 3, 1))
.show_matrix(arrX, main = "Source (X)")
.show_matrix(arrY, main = "Target (Y)")
.show_matrix(result$Ts[[1]], main = "Transport Plan 1\n(rows)")
.show_matrix(result$Ts[[2]], main = "Transport Plan 2\n(columns)")

par(mfrow = oldpar)

Parameter Reference

Tips:

• Larger num.sample gives more accurate gradients but is slower
• Start with smaller num.iter (e.g., 50) for exploration, increase for final results
• Custom loss functions can be passed, e.g., loss = function(x, y) (x - y)^2 for squared error

What’s Next?

The next vignette (otTensor-2: Optimal Tensor Transport) reproduces the experiments from the original paper (Kerdoncuff 2022), demonstrating OTT under all six f configurations:

• OTT_1 — standard OT (order-1 tensors)
• OTT_12 — Co-OT (each dimension independent)
• OTT_11 — Gromov-Wasserstein-like (shared plan)
• OTT_111 — Triplets (order-3, single shared plan)
• OTT_123 — tri-Co-OT (order-3, all independent)
• OTT_112 — GW collections (mixed sharing)