  Energy-Based Models (EBMs) assign unnormalized log-probability to data
samples. This functionality has a variety of applications, such as sample
synthesis, data denoising, sample restoration, outlier detection, Bayesian
reasoning, and many more. But training of EBMs using standard maximum
likelihood is extremely slow because it requires sampling from the model
distribution. Score matching potentially alleviates this problem. In
particular, denoising score matching \citep{vincent2011connection} has been
successfully used to train EBMs. Using noisy data samples with one fixed noise
level, these models learn fast and yield good results in data denoising
\citep{saremi2019neural}. However, demonstrations of such models in high
quality sample synthesis of high dimensional data were lacking. Recently,
\citet{song2019generative} have shown that a generative model trained by
denoising score matching accomplishes excellent sample synthesis, when trained
with data samples corrupted with multiple levels of noise. Here we provide
analysis and empirical evidence showing that training with multiple noise
levels is necessary when the data dimension is high. Leveraging this insight,
we propose a novel EBM trained with multi-scale denoising score matching. Our
model exhibits data generation performance comparable to state-of-the-art
techniques such as GANs, and sets a new baseline for EBMs. The proposed model
also provides density information and performs well in an image inpainting
task.
