Description: Wasserstein distance, also known as transportation distance, is a metric used to compare probability distributions. Its significance lies in measuring the ‘distance’ needed to transform one distribution into another, considering the cost of moving the ‘mass’ from one distribution to the other. This metric is based on optimal transport theory, which seeks to minimize the cost of moving resources from one place to another. Unlike other distance metrics, such as Kullback-Leibler divergence, Wasserstein distance has the advantage of being sensitive to the geometry of distributions, making it particularly useful in contexts where the shape of the distributions is relevant. Additionally, it is continuous and has desirable properties that make it suitable for applications in machine learning and statistics. In the field of machine learning, Wasserstein distance has been used to improve the optimization of generative models, such as Generative Adversarial Networks (GANs), by providing a more robust measure of similarity between generated and real distributions. In summary, Wasserstein distance is a powerful tool for comparing distributions, with applications ranging from statistics to machine learning.
History: Wasserstein distance was introduced by the Russian mathematician Leonid V. Kantorovich in 1942, who developed the theory of optimal transport. However, the term ‘Wasserstein distance’ became more popular later, in the 1970s, thanks to contributions from other mathematicians such as G. Szegö and R. J. McCann. Over the years, Wasserstein distance has evolved and been integrated into various areas of mathematics and statistics, especially in data analysis and machine learning.
Uses: Wasserstein distance is used in various applications, including hyperparameter optimization in machine learning models, evaluating the quality of generative models, and in federated learning to measure the similarity between data distributions from different nodes. It is also applied in statistics to compare empirical distributions and in game theory to analyze transport strategies.
Examples: A practical example of Wasserstein distance can be found in Generative Adversarial Networks (GANs), where Wasserstein distance is used to improve training stability and the quality of generated images. Another example is its use in federated learning, where the distance between data distributions from different devices is measured to ensure that the global model is trained effectively. Additionally, in statistics, it can be used to compare income distributions across different regions.
