Optimal Transport and applications to Machine Learning (1 ECTS, SSD: MAT05)

Optimal Transport (OT) is a mathematical theory introduced by Gaspard Monge in 1781 to study the
optimal allocation of resources and goods. Its original formulation, known as the Monge formulation,
aims at finding the best way to transport a probability distribution in $\mathcal{P}(\mathbb{R}^n)$ to
another by minimizing a transport cost computed with respect to a given cost function $c$.
Mathematically, this consists in finding a map $T : \mathbb{R}^n \to \mathbb{R}^n$ that minimizes the
cost of moving mass from the first measure to the second one. This formulation is highly flexible, as
it encompasses discrete, semi-discrete, and continuous settings, and the cost $c$ can be chosen to
enforce specific properties, such as restricting transport to certain regions or promoting mass
concentration.

Due to its flexibility and mathematical rigor, the theory experienced substantial development in the
20th century and gained relevance in fields such as economics, urban planning, image processing, and
biology. More recently---especially in the last decade---optimal transport has become a powerful tool
in machine learning. Many modern algorithms rely on estimating distances between data distributions
efficiently and accurately, and OT provides a natural framework for comparing such distributions by
quantifying the cost of transporting one into the other. This perspective, combined with fast
numerical methods for computing OT (notably entropic regularization and the Sinkhorn algorithm), has
made OT central in the design of new generative models, in solving inverse problems, and in improving
the robustness of neural networks.

In this series of lectures, we will cover: (1) an introduction to the classical formulations of
optimal transport and the core theoretical results; (2) entropic regularization and the Sinkhorn
algorithm for efficient computation of regularized OT; and (3) connections between OT and machine
learning, with a focus on adversarial generative models based on optimal transport (such as WGAN and
WAE). Time permitting, we will also discuss OT-based approaches to inverse problems.

Schedule: 

Tue 16/12 Aula G, 8:30-10:30

Wed 17/12 Aula Alfa, 13:30-15:30

Thu 18/12 Aula T.05, 10:30-12:30 

Contact: Giandomenico Orlandi/ Giacomo Albi