Learning Latent Space Hierarchical EBM Diffusion Models

The evolution of deep generative models has seen significant progress, from early VAEs and GANs to more recent diffusion models. Hierarchical models like NVAE and HVAE introduced layered latent variables to better capture complex data structures, but relied on simple Gaussian priors, leading to the 'prior hole' problem where the prior cannot fully represent the posterior distribution. Energy-based models (EBMs) have been proposed to enhance expressivity, especially in low-dimensional latent spaces, but their sampling via MCMC remains computationally intensive and challenging in multi-layer settings. Recent diffusion models, such as DDPM, have demonstrated high-quality generation, but integrating them with hierarchical EBMs to address multi-modal and multi-scale issues is still an open challenge. Existing methods often struggle with maintaining hierarchical dependencies during sampling and training, limiting their effectiveness in capturing rich semantic structures.

The core challenge lies in effectively modeling and sampling from the complex, multi-modal energy landscapes of multi-layer latent variable models. Traditional Gaussian priors are insufficient for capturing the rich intra-layer relations, leading to the 'prior hole' problem. MCMC-based sampling methods like Langevin dynamics are computationally expensive and often fail to explore all relevant modes efficiently, especially across different scales and layers. This results in poor training stability, limited hierarchical feature learning, and suboptimal sample quality. Developing a scalable, stable, and expressive approach that preserves hierarchical dependencies while enabling efficient sampling remains a critical bottleneck.

The main innovation is the integration of diffusion probabilistic models with conditional EBMs in a hierarchical latent space. The approach constructs a sequence of conditional EBMs operating in a transformed˜u-space, which maintains hierarchical dependencies during the forward diffusion process. This method employs a multi-stage KL divergence minimization and Langevin sampling tailored for each diffusion step, smoothing the energy landscape and reducing multi-modality issues. Unlike previous single-layer or data-space diffusion models, this framework explicitly models inter-layer relations, enabling scalable training and sampling in high-dimensional, multi-scale latent spaces. The use of˜u-space and progressive denoising preserves hierarchical semantics and improves sample diversity and quality.

• �� Define a multi-layer hierarchical generator with conditional Gaussian priors for each layer.
• �� Transform the latent space into a unified˜u-space via invertible functions to preserve hierarchical structure.
• �� Implement a diffusion process in˜u-space, gradually adding noise with a schedule designed to maintain layer dependencies.
• �� Construct a sequence of conditional EBMs at each diffusion step, modeling the energy of perturbed samples conditioned on the next step.
• �� Minimize the KL divergence between the forward diffusion trajectory and the EBM prior using Langevin dynamics for sampling.
• �� During training, update EBM parameters based on the difference in energy scores between perturbed and prior samples.
• �� For generation, reverse the diffusion process to reconstruct hierarchical latent variables, then decode to generate images.
• �� Ablation studies confirm the effectiveness of˜u-space and conditional EBMs in stabilizing training and improving sample quality.

The experiments utilize CIFAR-10, CelebA-HQ-256, and LSUN-Church-64 datasets, evaluating sample quality via FID and IS metrics. Baselines include NVAE, Joint-EBM, and diffusion recovery models. Hyperparameters such as diffusion steps T, Langevin steps, and learning rates are tuned for optimal performance. Ablation studies compare models with and without˜u-space, demonstrating significant improvements in sampling stability and quality. The hierarchical feature control is validated through latent space manipulations, showing meaningful semantic and detail variations. Quantitative results show the proposed model achieves FID scores of 8.78 on CIFAR-10, outperforming baselines, and similar gains on other datasets.

The proposed model achieves FID scores of 8.78 on CIFAR-10, 8.78 on CelebA-HQ-256, and 7.34 on LSUN-Church-64, surpassing existing methods like NVAE and Joint-EBM. It demonstrates superior hierarchical feature disentanglement, controllability, and sample diversity. Ablation results confirm that the˜u-space and conditional EBMs significantly reduce sampling difficulty, leading to more stable training and higher-quality samples. The hierarchical sampling experiments visually verify the model’s ability to manipulate semantic and detailed features at different layers, validating the effectiveness of the layered latent structure.

This framework is applicable to high-fidelity image synthesis, controllable content generation, and hierarchical feature learning. It can be integrated into virtual avatar creation, scene generation, and artistic design, especially where multi-level semantic control is desired. The approach also opens avenues for multi-modal data modeling, such as text-to-image synthesis, by capturing rich hierarchical representations. Its scalability and stability make it suitable for industrial deployment in content creation pipelines, virtual reality, and augmented reality applications.

Despite its strengths, the method involves high computational costs due to iterative diffusion and Langevin sampling, limiting real-time applications. Sensitivity to hyperparameters like diffusion schedule and noise levels requires careful tuning. The current validation is primarily on static image datasets; extending to videos or sequential data introduces additional complexity. Further work is needed to optimize efficiency, generalize across modalities, and reduce training time while maintaining high sample fidelity.