For NYT, Siobhan Roberts spoke to Alessio Figalli [archived link], a Fields Medal-winning mathematician and expert in optimal transport. They discussed how optimal transport has uses in everything from running a bakery to nature and AI:
Filed under: algorithms geometry machine learningOne challenge is using optimal transport in machine learning.
From a theoretical viewpoint, machine learning is just an optimization problem where you have a system, and you want to optimize some parameters, or features, so that the machine will do a certain number of tasks.
To classify images, optimal transport measures how similar two images are by comparing features like colors or textures and putting these features into alignment — transporting them — between the two images. This technique helps improve accuracy, making models more robust to changes or distortions.
These are very high-dimensional phenomena. You are trying to understand objects that have many features, many parameters, and every feature corresponds to one dimension. So if you have 50 features, you are in 50-dimensional space.
The higher the dimension where the object lives, the more complex the optimal transport problem is — it requires too much time, too much data to solve the problem, and you will never be able to do it. This is called the curse of dimensionality. Recently people have been trying to look at ways to avoid the curse of dimensionality. One idea is to develop a new type of optimal transport.
Alessio Figalli