In this talk, I will discuss our recent theoretical advancements in generative modeling. The first part of the presentation will focus on learning distributions with symmetry. I will introduce results on the sample complexity of empirical estimations of probability divergences for group-invariant distributions, and present performance guarantees for GANs and score-based generative models that incorporate symmetry. Notably, I will offer the first quantitative comparison between data augmentation and directly embedding symmetry into models, highlighting the latter as a more fundamental approach for efficient learning. These findings underscore how incorporating symmetry into generative models can significantly enhance learning efficiency, particularly in data-limited scenarios. The second part will cover $\alpha$-divergences with Wasserstein-1 regularization. These divergences can be interpreted as $\alpha$-divergences constrained to Lipschitz test functions in their variational form. I will demonstrate how generative learning can be made agnostic to assumptions about target distributions, including those with heavy tails or low-dimensional and fractal supports, through the use of these divergences as objective functionals. I will outline the conditions for the finiteness of these divergences under minimal assumptions on the target distribution along with the gradient flow formulation associated with them. This framework provides guarantees for various machine learning algorithms that optimize over this class of divergences. 

In this talk, we will start by describing how classical tools from probability

offer a robust framework to understand the dynamics of waves via appropriate ensembles

on phase space rather than particular microscopic dynamical trajectories. We will continue

by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this

context and how it opened the door to unveil the random structures of nonlinear waves that

live on high frequencies and fine scales as they propagate. We will then discuss how these 

ideas broke the logjam in the study of the Gibbs measures associated to nonlinear 

Schrödinger equations in the context of equilibrium statistical mechanics and of the 

hyperbolic $\Phi^4_3$ model in the context of constructive quantum field theory. 

We will end with some open challenges about the long-time propagation of randomness 

and out-of-equilibrium dynamics.

Please join us for pre-seminar Tea

I will describe a joint project with Tobias Ekholm, Pietro Longhi, and Vivek Shende constructing a map from the HOMFLYPT skein module of a 3-manifold M to that of its branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M. The map is defined by counting holomorphic curves and is a vast generalization of the quantum UV-IR map of Neitzke and Yan, which is a close cousin of the quantum trace map of Bonahon and Wong. The existence of this map has some interesting consequences in the theory of skein-valued curve counts, and I will discuss some of them if time permits.