Investigating noise distributions beyond Gaussian in diffusion generative models remains an open challenge. The Gaussian case has been a large success experimentally and theoretically, admitting a unified stochastic differential equation (SDE) framework, encompassing score-based and denoising formulations. Recent studies have investigated the potential of \emph{heavy-tailed} noise distributions to mitigate mode collapse and effectively manage datasets exhibiting class imbalance, heavy tails, or prominent outliers. Very recently, Yoon et al.\ (NeurIPS 2023), presented the Levy-Ito model (LIM), directly extending the SDE-based framework to a class of heavy-tailed SDEs, where the injected noise followed an $\alpha$-stable distribution -- a rich class of heavy-tailed distributions. Despite its theoretical elegance and performance improvements, LIM relies on highly involved mathematical techniques, which may limit its accessibility and hinder its broader adoption and further development. In this study, we take a step back, and instead of starting from the SDE formulation, we extend the denoising diffusion probabilistic model (DDPM) by directly replacing the Gaussian noise with $\alpha$-stable noise. By using only elementary proof techniques, we show that the proposed approach, \emph{denoising L\'{e}vy probabilistic model} (DLPM) algorithmically boils down to running vanilla DDPM with minor modifications, hence allowing the use of existing implementations with minimal changes. Remarkably, as opposed to the Gaussian case, DLPM and LIM yield different training algorithms and different backward processes, leading to distinct sampling algorithms. This fundamental difference translates favorably for the performance of DLPM in various aspects: our experiments show that DLPM achieves better coverage of the tails of the data distribution, better generation of unbalanced datasets, and improved computation times requiring significantly smaller number of backward steps.

This paper introduces the Discrete Markov Probabilistic Model (DMPM), a novel algorithm for discrete data generation. The algorithm operates in discrete space, where the noising process is a continuous-time Markov chain that can be sampled exactly via a Poissonian clock that flips labels uniformly at random. The time-reversal process, like the forward noise process, is a jump process, with its intensity governed by a discrete analogue of the classical score function. Crucially, this intensity is proven to be the conditional expectation of a function of the forward process, strengthening its theoretical alignment with score-based generative models while ensuring robustness and efficiency. We further establish convergence bounds for the algorithm under minimal assumptions and demonstrate its effectiveness through experiments on low-dimensional Bernoulli-distributed datasets and high-dimensional binary MNIST data. The results highlight its strong performance in generating discrete structures. This work bridges theoretical foundations and practical applications, advancing the development of effective and theoretically grounded discrete generative modeling.

Score-based generative models (SGMs) have emerged as one of the most popular classes of generative models. A substantial body of work now exists on the analysis of SGMs, focusing either on discretization aspects or on their statistical performance. In the latter case, bounds have been derived, under various metrics, between the true data distribution and the distribution induced by the SGM, often demonstrating polynomial convergence rates with respect to the number of training samples. However, these approaches adopt a largely approximation theory viewpoint, which tends to be overly pessimistic and relatively coarse. In particular, they fail to fully explain the empirical success of SGMs or capture the role of the optimization algorithm used in practice to train the score network. To support this observation, we first present simple experiments illustrating the concrete impact of optimization hyperparameters on the generalization ability of the generated distribution. Then, this paper aims to bridge this theoretical gap by providing the first algorithmic- and data-dependent generalization analysis for SGMs. In particular, we establish bounds that explicitly account for the optimization dynamics of the learning algorithm, offering new insights into the generalization behavior of SGMs. Our theoretical findings are supported by empirical results on several datasets.

We introduce a novel class of generative models based on piecewise deterministic Markov processes (PDMPs), a family of non-diffusive stochastic processes consisting of deterministic motion and random jumps at random times. Similarly to diffusions, such Markov processes admit time reversals that turn out to be PDMPs as well. We apply this observation to three PDMPs considered in the literature: the Zig-Zag process, Bouncy Particle Sampler, and Randomised Hamiltonian Monte Carlo. For these three particular instances, we show that the jump rates and kernels of the corresponding time reversals admit explicit expressions depending on some conditional densities of the PDMP under consideration before and after a jump. Based on these results, we propose efficient training procedures to learn these characteristics and consider methods to approximately simulate the reverse process. Finally, we provide bounds in the total variation distance between the data distribution and the resulting distribution of our model in the case where the base distribution is the standard $d$-dimensional Gaussian distribution. Promising numerical simulations support further investigations into this class of models.