  Maximum likelihood (ML) learning for energy-based models (EBMs) is
challenging, partly due to non-convergence of Markov chain Monte Carlo.Several
variations of ML learning have been proposed, but existing methods all fail to
achieve both post-training image generation and proper density estimation. We
propose to introduce diffusion data and learn a joint EBM, called diffusion
assisted-EBMs, through persistent training (i.e., using persistent contrastive
divergence) with an enhanced sampling algorithm to properly sample from
complex, multimodal distributions. We present results from a 2D illustrative
experiment and image experiments and demonstrate that, for the first time for
image data, persistently trained EBMs can {\it simultaneously} achieve long-run
stability, post-training image generation, and superior out-of-distribution
detection.
