Maximum Likelihood Training of Implicit Nonlinear Diffusion Models

Whereas diverse variations of diffusion models exist, expanding the linear diffusion into a nonlinear diffusion process is investigated only by a few works. The nonlinearity effect has been hardly understood, but intuitively, there would be more promising diffusion patterns to optimally train the generative distribution towards the data distribution. This paper introduces such a data-adaptive and nonlinear diffusion process for score-based diffusion models. The proposed Implicit Nonlinear Diffusion Model (INDM) learns the nonlinear diffusion process by combining a normalizing flow and a diffusion process. Specifically, INDM implicitly constructs a nonlinear diffusion on the \textit{data space} by leveraging a linear diffusion on the \textit{latent space} through a flow network. This flow network is the key to forming a nonlinear diffusion as the nonlinearity fully depends on the flow network. This flexible nonlinearity is what improves the learning curve of INDM to nearly MLE training, compared against the non-MLE training of DDPM++, which turns out to be a special case of INDM with the identity flow. Also, training the nonlinear diffusion empirically yields a sampling-friendly latent diffusion that the sample trajectory of INDM is closer to an optimal transport than the trajectories of previous research. In experiments, INDM achieves the state-of-the-art FID on CelebA.