Learning Energy-Based Models in High-Dimensional Spaces with Multiscale Denoising-Score Matching.

Energy-based models (EBMs) assign an 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, the 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 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. However, demonstrations of such models in the high-quality sample synthesis of high-dimensional data were lacking. Recently, a paper showed 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 an 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 multiscale denoising-score matching. Our model exhibits a 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 on an image-inpainting task.