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Diffusion models are a new class of generative models that revolve around theestimation of the score function associated with a stochastic differentialequation. Subsequent to its acquisition, the approximated score function isthen harnessed to simulate the corresponding time-reversal process, ultimatelyenabling the generation of approximate data samples. Despite their evidentpractical significance these models carry, a notable challenge persists in theform of a lack of comprehensive quantitative results, especially in scenariosinvolving non-regular scores and estimators. In almost all reported bounds inKullback Leibler (KL) divergence, it is assumed that either the score functionor its approximation is Lipschitz uniformly in time. However, this condition isvery restrictive in practice or appears to be difficult to establish. Tocircumvent this issue, previous works mainly focused on establishingconvergence bounds in KL for an early stopped version of the diffusion modeland a smoothed version of the data distribution, or assuming that the datadistribution is supported on a compact manifold. These explorations have led tointeresting bounds in either Wasserstein or Fortet-Mourier metrics. However,the question remains about the relevance of such early-stopping procedure orcompactness conditions. In particular, if there exist a natural and mildcondition ensuring explicit and sharp convergence bounds in KL. In thisarticle, we tackle the aforementioned limitations by focusing on scorediffusion models with fixed step size stemming from the Ornstein-Uhlenbecksemigroup and its kinetic counterpart. Our study provides a rigorous analysis,yielding simple, improved and sharp convergence bounds in KL applicable to anydata distribution with finite Fisher information with respect to the standardGaussian distribution.

KL Convergence Guarantees for Score diffusion models under minimal data assumptions