Seminários e cursos curtos

Na terceira sessão serão abordados os seguintes tópicos de análise $p$-ádica:

• Breve sumário sobre a topologia p-ádica
• Breve resumo sobre séries $p$-adicas
• Funções continuas e diferenciabilidade

The development of the theory of martingale Hardy spaces has been strongly influenced by the classical theory. Because of this it is inevitable to compare results of this theory to those on classical Hardy spaces and during the decades this new direction was developing in this way and many similarities between these theories have already been found. But nowadays a lot of new results were obtained in the theory of martingale Hardy spaces which are new in classical case.

This lecture is devoted to review theory of martingale Hardy spaces. In particular, we give atomic decomposition of these spaces and show how this theorem simplify the proofs of boundedness of any sub-linear operators on these spaces. For the illustration of bounded operators on the martingale Hardy spaces we define Vilenkin groups and their characters which are called Vilenkin functions. We also define Fourier series with respect to Vilenkin system and maximal operators related to these operators. Moreover, we define modulus of continuity in martingale Hardy spaces and derive necessary and sufficient conditions in the terms of modulus of continuity such that partial sums with respect to one Vilenkin-Fourier series converge in $H^p$ norm. We also investigate classical Hardy spaces, modulus of continuity in these spaces and present one to one analogues of these results for partial sums in this classical case.

At the end of this presentation we state some open problem how to generate these two similar results for Hardy spaces on some locally compact Abelian groups and partial sums with respect to character functions of these groups. Moreover, we also state similar open problems for variable martingale Hardy spaces. Finally, we will choose opposite approach and will try to state and investigate results for partial sums of the two-dimensional Fourier series in martingale Hardy spaces for classical case of Hardy spaces.

Diffusion probabilistic models have become the state-of-the-art tool in generative methods, used to generate high-resolution samples from very high-dimension distributions (e.g. images). Although very effective, they suffer some drawbacks:

1. as opposed to variational encoders, the dimension of the problem remains high during the generation process and
2. they can be prone to memorization of the training dataset.

In this talk, we first provide an introduction to generative modeling, with a focus on diffusion models from the point of view of stochastic PDEs. Then, we introduce a kernel-smoothed empirical score and study the bias-variance of this estimator. We find improved bounds on the KL-divergence between a true measure and an approximate measure generated by using the smoothed empirical score. This score estimator leads to less memorization and better generalization. We demonstrate these findings on synthetic and real datasets, combining diffusion models with variational encoders to reduce the dimensionality of the problem.