Stochastic analysis and rough paths methods in machine learning

The PIs research is concerned with advanced probabilistic models which have potential real-world applications in cutting-edge machine learning techniques. It aims to bring mathematical rigor and come up with new methods related to complex systems in image processing, reinforcement learning, and generative AI. Invoking recent breakthroughs in stochastic analysis as well as developing new tools, the project intends to make progress in the following directions: it introduces new ways to extract meaningful features from images, possibly enhances decision-making systems through reinforcement learning in random environments, and improves the theoretical understanding of generative models such as diffusion-based algorithms. These developments have the potential to contribute to more interpretable, robust, and effective AI systems, with applications ranging from medical imaging to autonomous driving. Beyond technical contributions, the work promotes interdisciplinary collaboration and offers strong mentorship opportunities for students and junior researchers. On a technical level, the project explores four main directions: (1) the development of 2D-signatures based on rough paths and Hopf algebra structures to extract robust features from image data; (2) the construction of new image descriptors via expansions inspired by regularity structures and nonlinear PDEs; (3) the formulation of reinforcement learning problems as relaxed control problems driven by rough paths, with entropy regularization and rigorous optimization procedures for the value function; and (4) the study of generative modeling through reversed diffusions and score-based methods, with a focus on improving theoretical guarantees and algorithmic implementations using Malliavin calculus. Each of these threads addresses fundamental challenges in modeling, analysis, and numerical approximation, and together they aim to bridge the gap between abstract mathematical theory and practical machine learning tools. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.