  Diffusion-based generative models have demonstrated a capacity for
perceptually impressive synthesis, but can they also be great likelihood-based
models? We answer this in the affirmative, and introduce a family of
diffusion-based generative models that obtain state-of-the-art likelihoods on
standard image density estimation benchmarks. Unlike other diffusion-based
models, our method allows for efficient optimization of the noise schedule
jointly with the rest of the model. We show that the variational lower bound
(VLB) simplifies to a remarkably short expression in terms of the
signal-to-noise ratio of the diffused data, thereby improving our theoretical
understanding of this model class. Using this insight, we prove an equivalence
between several models proposed in the literature. In addition, we show that
the continuous-time VLB is invariant to the noise schedule, except for the
signal-to-noise ratio at its endpoints. This enables us to learn a noise
schedule that minimizes the variance of the resulting VLB estimator, leading to
faster optimization. Combining these advances with architectural improvements,
we obtain state-of-the-art likelihoods on image density estimation benchmarks,
outperforming autoregressive models that have dominated these benchmarks for
many years, with often significantly faster optimization. In addition, we show
how to use the model as part of a bits-back compression scheme, and demonstrate
lossless compression rates close to the theoretical optimum. Code is available
at https://github.com/google-research/vdm .
