In this note, we examine the forward-Euler discretization for simulating Wasserstein gradient flows. We provide two counter-examples showcasing the failure of this discretization even for a simple case where the energy functional is defined as the KL divergence against some nicely structured probability densities. A simple explanation of this failure is also discussed.

The number of sampling methods could be daunting for a practitioner looking to cast powerful machine learning methods to their specific problem. This paper takes a theoretical stance to review and organize many sampling approaches in the ``generative modeling'' setting, where one wants to generate new data that are similar to some training examples. By revealing links between existing methods, it might prove useful to overcome some of the current challenges in sampling with diffusion models, such as long inference time due to diffusion simulation, or the lack of diversity in generated samples.

Generative Adversarial Networks (GAN) are powerful methods for learning probability measures and performing realistic sampling [21]. Algorithms in this class aim to reproduce the sampling behavior of the target distribution, rather than explicitly fitting a density function. This is done by modeling the target probability as the pushforward of a probability measure in a latent space. Since their introduction, GANs have achieved remarkable progress. From a practical perspective, a large number of model architectures have been explored, leading to impressive results in data generation [48, 24, 26]. From the side of theory, attention has been devoted to identify rich metrics for generator training, such as f-divergences [36], integral probability metrics (IPM) [16] or optimal transport distances [3], as well as studying their approximation properties [28, 5, 51]. From the statistical perspective, progression has been slower. While recent work has set the first steps towards a characterization of the generalization properties of GANs with IPM loss functions [4, 51, 27, 41, 45], a full theoretical understanding of the main building blocks of the GAN framework is still missing. In this work, we focus on optimal transport-based loss functions [20] and study the impact of two key quantities of the GANs paradigm on the overall generalization performance.