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Maximum likelihood (ML) learning for energy-based models (EBMs) ischallenging, partly due to non-convergence of Markov chain Monte Carlo.Severalvariations of ML learning have been proposed, but existing methods all fail toachieve both post-training image generation and proper density estimation. Wepropose to introduce diffusion data and learn a joint EBM, called diffusionassisted-EBMs, through persistent training (i.e., using persistent contrastivedivergence) with an enhanced sampling algorithm to properly sample fromcomplex, multimodal distributions. We present results from a 2D illustrativeexperiment and image experiments and demonstrate that, for the first time forimage data, persistently trained EBMs can {\it simultaneously} achieve long-runstability, post-training image generation, and superior out-of-distributiondetection.

Persistently Trained, Diffusion-assisted Energy-based Models