  Score-based generative models are shown to achieve remarkable empirical
performances in various applications such as image generation and audio
synthesis. However, a theoretical understanding of score-based diffusion models
is still incomplete. Recently, Song et al. showed that the training objective
of score-based generative models is equivalent to minimizing the
Kullback-Leibler divergence of the generated distribution from the data
distribution. In this work, we show that score-based models also minimize the
Wasserstein distance between them under suitable assumptions on the model.
Specifically, we prove that the Wasserstein distance is upper bounded by the
square root of the objective function up to multiplicative constants and a
fixed constant offset. Our proof is based on a novel application of the theory
of optimal transport, which can be of independent interest to the society. Our
numerical experiments support our findings. By analyzing our upper bounds, we
provide a few techniques to obtain tighter upper bounds.
