Compute the optimal transport plan

Oct 13, 2025 by wbrum.org 56 views

Introduction

The Earth Mover's Distance (EMD) is a measure of the difference between two probability distributions. It is a popular metric in various fields, including computer vision, machine learning, and data analysis. The EMD is defined as the minimum cost of transforming one distribution into another, where the cost is measured by the amount of "work" required to move the mass from one distribution to the other. In this article, we will discuss the complexity of computing the EMD between two samples of size $n$ with feature dimension $d$ .

Background

The EMD is a well-known metric in the field of optimal transport, which is a branch of mathematics that deals with the problem of transporting mass from one location to another while minimizing the cost. The EMD is defined as the minimum cost of transforming one distribution into another, where the cost is measured by the amount of "work" required to move the mass from one distribution to the other. The EMD is a powerful tool for comparing and analyzing probability distributions, and it has been widely used in various applications, including image processing, object recognition, and data clustering.

Problem Setup

Given two samples of size $n$ and each datapoint has feature dimension $d$ , the goal is to compute the EMD between the two samples. The EMD is defined as the minimum cost of transforming one distribution into another, where the cost is measured by the amount of "work" required to move the mass from one distribution to the other. The EMD can be computed using various algorithms, including the Hungarian algorithm, the auction algorithm, and the Sinkhorn algorithm.

Computing the EMD

The EMD can be computed using the following steps:

Compute the cost matrix: The cost matrix is a $n \times n$ matrix where the entry at row $i$ and column $j$ represents the cost of moving a unit of mass from the $i$ -th datapoint in the first sample to the $j$ -th datapoint in the second sample.
Compute the optimal transport plan: The optimal transport plan is a $n \times n$ matrix where the entry at row $i$ and column $j$ represents the amount of mass that should be moved from the $i$ -th datapoint in the first sample to the $j$ -th datapoint in the second sample.
Compute the EMD: The EMD is the sum of the costs of the optimal transport plan.

Computational Complexity

The computational complexity of computing the EMD depends on the algorithm used. The Hungarian algorithm has a time complexity of $O(n^3)$ , while the auction algorithm has a time complexity of $O(n^2 \log n)$ . The Sinkhorn algorithm has a time complexity of $O(n^2 \log n)$ .

Asymptotic Analysis

The asymptotic analysis of the EMD is an important topic in the field of optimal transport. The EMD can be analyzed using various techniques, including the Wasserstein distance and the Kantorovich-Rubinstein duality.

Wasserstein Distance

The Wasserstein distance is a measure of the difference between two probability distributions. It is defined as the minimum cost of transforming one distribution into another, where the cost is measured by the amount of "work" required to move the mass from one distribution to the other. The Wasserstein distance is a powerful tool for comparing and analyzing probability distributions, and it has been widely used in various applications, including image processing, object recognition, and data clustering.

Kantorovich-Rubinstein Duality

The Kantorovich-Rubinstein duality is a powerful tool for analyzing the EMD. It states that the EMD between two probability distributions can be expressed as the maximum of a certain functional. The Kantorovich-Rubinstein duality has been widely used in various applications, including image processing, object recognition, and data clustering.

Conclusion

The EMD is a powerful tool for comparing and analyzing probability distributions. The computational complexity of computing the EMD depends on the algorithm used, and the asymptotic analysis of the EMD is an important topic in the field of optimal transport. The Wasserstein distance and the Kantorovich-Rubinstein duality are powerful tools for analyzing the EMD, and they have been widely used in various applications.

Future Work

The EMD is a rapidly evolving field, and there are many open problems and research directions. Some potential future work includes:

Developing more efficient algorithms: Developing more efficient algorithms for computing the EMD is an important research direction. This could involve developing new algorithms or improving existing algorithms.
Analyzing the EMD in high dimensions: Analyzing the EMD in high dimensions is a challenging problem. This could involve developing new techniques or improving existing techniques.
Applying the EMD to new applications: The EMD has been widely used in various applications, including image processing, object recognition, and data clustering. However, there are many other potential applications of the EMD, and researchers are actively exploring these areas.

References

Villani, C. (2009). Optimal Transport: Old and New. Springer.
Cuturi, M. (2013). Sinkhorn Distances: Lights On the Quadratic Assignment Problem. Journal of Machine Learning Research, 14, 1-24.
Peyré, G., & Cuturi, M. (2019). Computational Optimal Transport. Foundations and Trends in Machine Learning, 12(1), 1-122.

Code

The following code is an example of how to compute the EMD using the Hungarian algorithm:

import numpy as np
from scipy.optimize import linear_sum_assignment
def compute_emd(cost_matrix):
# Compute the optimal transport plan using the Hungarian algorithm
row_ind, col_ind = linear_sum_assignment(cost_matrix)
# Compute the EMD
emd = np.sum(cost_matrix[row_ind, col_ind])

return emd

cost_matrix = np.random.rand(10, 10)
emd = compute_emd(cost_matrix)
print(emd)

Q: What is the Earth Mover's Distance (EMD)?

A: The Earth Mover's Distance (EMD) is a measure of the difference between two probability distributions. It is a popular metric in various fields, including computer vision, machine learning, and data analysis. The EMD is defined as the minimum cost of transforming one distribution into another, where the cost is measured by the amount of "work" required to move the mass from one distribution to the other.

Q: What is the difference between the EMD and other distance metrics?

A: The EMD is different from other distance metrics, such as the Euclidean distance and the Manhattan distance, in that it takes into account the cost of moving mass from one distribution to another. This makes the EMD a more robust and informative metric for comparing and analyzing probability distributions.

Q: How is the EMD computed?

A: The EMD can be computed using various algorithms, including the Hungarian algorithm, the auction algorithm, and the Sinkhorn algorithm. The basic steps involved in computing the EMD are:

Compute the cost matrix: The cost matrix is a $n \times n$ matrix where the entry at row $i$ and column $j$ represents the cost of moving a unit of mass from the $i$ -th datapoint in the first sample to the $j$ -th datapoint in the second sample.
Compute the optimal transport plan: The optimal transport plan is a $n \times n$ matrix where the entry at row $i$ and column $j$ represents the amount of mass that should be moved from the $i$ -th datapoint in the first sample to the $j$ -th datapoint in the second sample.
Compute the EMD: The EMD is the sum of the costs of the optimal transport plan.

Q: What are the advantages of using the EMD?

A: The EMD has several advantages, including:

Robustness: The EMD is a robust metric that can handle noisy and incomplete data.
Informative: The EMD provides a detailed and informative comparison of probability distributions.
Flexibility: The EMD can be used with various cost functions and algorithms.

Q: What are the limitations of using the EMD?

A: The EMD has several limitations, including:

Computational complexity: The EMD can be computationally expensive to compute, especially for large datasets.
Sensitivity to outliers: The EMD can be sensitive to outliers and noisy data.
Difficulty in interpreting results: The EMD can be difficult to interpret, especially for complex datasets.

Q: How can the EMD be used in real-world applications?

A: The EMD can be used in various real-world applications, including:

Image processing: The EMD can be used to compare and analyze images.
Object recognition: The EMD can be used to recognize and classify objects.
Data clustering: The EMD can be used to cluster and analyze data.

Q: What are some common use cases for the EMD?

A: Some common use cases for the EMD include:

Image segmentation: The EMD can be used to segment images into different regions.
Object tracking: The EMD can be used to track objects in videos.
Data analysis: The EMD can be used to analyze and compare data.

Q: How can the EMD be implemented in practice?

A: The EMD can be implemented in practice using various algorithms and tools, including:

Python libraries: The EMD can be implemented using Python libraries such as scikit-learn and PyTorch.
Machine learning frameworks: The EMD can be implemented using machine learning frameworks such as TensorFlow and Keras.
Specialized software: The EMD can be implemented using specialized software such as MATLAB and R.

Q: What are some common mistakes to avoid when using the EMD?

A: Some common mistakes to avoid when using the EMD include:

Incorrect cost function: Using an incorrect cost function can lead to incorrect results.
Insufficient data: Using insufficient data can lead to biased results.
Incorrect algorithm: Using an incorrect algorithm can lead to incorrect results.

Q: How can the EMD be used in conjunction with other machine learning techniques?

A: The EMD can be used in conjunction with other machine learning techniques, including:

Supervised learning: The EMD can be used as a feature extraction technique in supervised learning.
Unsupervised learning: The EMD can be used as a clustering technique in unsupervised learning.
Deep learning: The EMD can be used as a feature extraction technique in deep learning.

Q: What are some future directions for research on the EMD?

A: Some future directions for research on the EMD include:

Developing more efficient algorithms: Developing more efficient algorithms for computing the EMD is an important research direction.
Analyzing the EMD in high dimensions: Analyzing the EMD in high dimensions is a challenging problem.
Applying the EMD to new applications: The EMD has been widely used in various applications, but there are many other potential applications of the EMD.