Latent space Energy-Based Models (EBMs), also known as energy-based priors, have drawn growing interests in the field of generative modeling due to its flexibility in the formulation and strong modeling power of the latent space. However, the common practice of learning latent space EBMs with non-convergent short-run MCMC for prior and posterior sampling is hindering the model from further progress; the degenerate MCMC sampling quality in practice often leads to degraded generation quality and instability in training, especially with highly multi-modal and/or high-dimensional target distributions. To remedy this sampling issue, in this paper we introduce a simple but effective diffusion-based amortization method for long-run MCMC sampling and develop a novel learning algorithm for the latent space EBM based on it. We provide theoretical evidence that the learned amortization of MCMC is a valid long-run MCMC sampler.

We show additional image synthesis in Fig.2. For reported numbers in main text, we adopt the network structure that contains Residue Blocks (see implementation details in Tab.5). We then test our model for the task of image inpainting. As shown in Fig.1, our This is the marginal version of Eqn.8 shown in the main text. 2 2.3 Learning Algorithm Three models are trained in an alternative and iterative manner based on the current model parameters. Compared to Eqn.3 and Eqn.6 in the main text, Eqn.5 and Eqn.6 start with initial points initialized We present the learning algorithm in Alg.1.

In this appendix, we provide pseudo-code algorithms explaining how to build the metric from a29 trained VAEandhowtousetheproposed sampling process. B.2.1 TheHMCsampler43 In the sampling process we propose to rely on the Hamiltonian Monte Carlo sampler to sample44 fromtheRiemanian uniformdistribution. Moreover,sinceG(z)issmooth andhas66 a closed form, it can be differentiated with respect toz pretty easily. Figure 5: Closest element inthetraining set(Near.) Each model is trained on each label of the train set and used to generate 2k samples per89 class.

Latent space Energy-Based Models (EBMs), also known as energy-based priors, have drawn growing interests in the field of generative modeling due to its flexibility in the formulation and strong modeling power of the latent space. However, the common practice of learning latent space EBMs with non-convergent short-run MCMC for prior and posterior sampling is hindering the model from further progress; the degenerate MCMC sampling quality in practice often leads to degraded generation quality and instability in training, especially with highly multi-modal and/or high-dimensional target distributions. To remedy this sampling issue, in this paper we introduce a simple but effective diffusion-based amortization method for long-run MCMC sampling and develop a novel learning algorithm for the latent space EBM based on it. We provide theoretical evidence that the learned amortization of MCMC is a valid long-run MCMC sampler.

We show additional image synthesis in Fig.2. For reported numbers in main text, we adopt the network structure that contains Residue Blocks (see implementation details in Tab.5). We then test our model for the task of image inpainting. As shown in Fig.1, our This is the marginal version of Eqn.8 shown in the main text. 2 2.3 Learning Algorithm Three models are trained in an alternative and iterative manner based on the current model parameters. Compared to Eqn.3 and Eqn.6 in the main text, Eqn.5 and Eqn.6 start with initial points initialized We present the learning algorithm in Alg.1.