Description: Optimal transport is a mathematical theory that focuses on comparing probability distributions, seeking the most efficient way to move resources from a set of source points to a set of destination points. This theory is based on minimizing costs associated with transportation, considering both the amount of resources to be moved and the distances between points. In the context of machine learning, optimal transport is used to measure the similarity between two probability distributions, facilitating the evaluation of the quality of samples generated by models compared to real samples. Through techniques such as Wasserstein distance, optimal transport allows for more stable and effective convergence in the training of generative models, overcoming some limitations of traditional metrics. This methodology not only improves the quality of generated images but also provides a solid theoretical framework for understanding how data is distributed in feature space. In summary, optimal transport is a powerful tool in mathematics and machine learning that optimizes processes of comparison and data generation.
History: The concept of optimal transport dates back to the work of Gaspard Monge in 1781, who formulated the transportation problem in the context of economics and logistics. Throughout the 20th century, the theory was developed and formalized by mathematicians such as Leonid Kantorovich, who introduced the linear programming approach to solve transportation problems. In recent decades, optimal transport has found applications in various fields, including probability theory and machine learning, especially in the context of generative models.
Uses: Optimal transport is used in various applications, such as logistics to optimize delivery routes, in economics to model resource distribution, and in statistics to compare probability distributions. In the field of machine learning, it is applied in the evaluation of generative models, improving the quality of generated samples and facilitating convergence in the training of neural networks.
Examples: A practical example of the use of optimal transport is the implementation of Wasserstein distance in generative models, which allows for evaluating the quality of generated images compared to a real dataset. Another case is the optimization of distribution routes in logistics companies, where the goal is to minimize transportation costs between multiple source and destination points.
