Title:

Abstract:

On Learning, robustness, and the geometry of noise

Suppose you are given a set of data points that are sampled from an unknown distribution in a set of reasonable probabilistic models. How do you construct an estimator, i.e. a function of your observations, for a given quantity of interest in an “optimal” way? This is a question that statisticians have asked for a long time and that in contemporary data sciences has become much more relevant as new notions of optimality beyond accuracy such as robustness, fairness, and privacy have gained relevance. In this lecture series, I will discuss this question, revisiting some of the classical notions of optimality of estimators and then focusing on the robustness of estimators to data perturbations. In contrast to a more standard statistical course on the topic, in this lecture series the emphasis will be on the geometry of statistical inference. This perspective will lead us to study different geometries in the space of probability measures and use their structures to link them to different noise models and to the corresponding notions of robustness of estimators to those noise models.

Title:

Abstract:

Machine Learning from an Applied Mathematician’s Perspective

Machine Learning has emerged as one of the most transformative forces in contemporary science and technology. In this three-lecture series, I will discuss Machine Learning through the lens of applied mathematics, emphasizing its connections with control theory, partial differential equations, and numerical analysis. 

In the first lecture, we will revisit the historical and conceptual links between Machine Learning and Systems Control (Cybernetics). This point of view allows us to reinterpret representation and expressivity properties of deep neural networks in terms of ensemble or simultaneous controllability of neural differential equations.  

The second lecture will focus on the use of neural network architectures as numerical approximation tools. We will consider, as a guiding example, the classical Dirichlet problem for the Laplace equation, formulated via energy minimization under neural-network constraints. Particular attention will be paid to the lack of convexity and coercivity in the resulting optimization problems. We will show how relaxation techniques may restore convexity at the price of losing coercivity, and we will discuss the mathematical implications of this trade-off for analysis and computation. 

In the third lecture, we will present a PDE-based perspective on generative diffusion models. Their convergence can be reinterpreted in terms of the asymptotic behavior of Fokker–Planck equations driven by the so-called score vector field. We will explain how classical tools, such as Li-Yau-type differential inequalities for positive solutions of the heat equation, provide insight into the regularization and convergence properties of these models. 

The series will conclude with a discussion of open problems and promising directions for future research at the interface of control theory, PDEs, numerical analysis, and modern Machine Learning.