Diffusion Probabilistic Fields (2303.00165v1)

Abstract: Diffusion probabilistic models have quickly become a major approach for generative modeling of images, 3D geometry, video and other domains. However, to adapt diffusion generative modeling to these domains the denoising network needs to be carefully designed for each domain independently, oftentimes under the assumption that data lives in a Euclidean grid. In this paper we introduce Diffusion Probabilistic Fields (DPF), a diffusion model that can learn distributions over continuous functions defined over metric spaces, commonly known as fields. We extend the formulation of diffusion probabilistic models to deal with this field parametrization in an explicit way, enabling us to define an end-to-end learning algorithm that side-steps the requirement of representing fields with latent vectors as in previous approaches (Dupont et al., 2022a; Du et al., 2021). We empirically show that, while using the same denoising network, DPF effectively deals with different modalities like 2D images and 3D geometry, in addition to modeling distributions over fields defined on non-Euclidean metric spaces.

• The paper introduces Diffusion Probabilistic Fields to model continuous functions, bypassing Euclidean limitations and unifying diverse data modalities.
• The paper demonstrates competitive performance on 2D images and 3D geometry tasks using metrics like FID and coverage against specialized models.
• The paper sets the stage for future research in efficient sampling and architectural optimization for generative modeling on non-Euclidean data.

Diffusion Probabilistic Fields: A Unified Approach to Generative Modeling

The paper introduces an innovative approach to generative modeling called Diffusion Probabilistic Fields (DPF). This method extends the application of diffusion probabilistic models to learn distributions over continuous functions, or fields, which are defined over metric spaces. The DPF framework stands out by enabling generative modeling across various domains—including 2D images, 3D geometry, and non-Euclidean spaces—without the necessity of designing domain-specific denoising networks.

Key Contributions

DPF addresses two central issues prevalent in current diffusion models:

1. Euclidean Assumptions: Traditional models often require data to reside within Euclidean grids. DPF circumvents this limitation by employing a field-based representation.
2. Domain-Specific Tuning: Custom architectures for differing data types (images, videos, 3D geometry) are typically required. DPF provides a unified architecture that efficiently handles diverse data modalities.

Methodology

The DPF model operates by interpreting data as continuous functions (fields) represented through context (coordinate-signal) pairs. This explicit parameterization allows for direct end-to-end training, avoiding the need for latent vector representations as seen in some prior techniques like Functa and GEM. By treating data as a field $f: \mathcal{M} \rightarrow \mathcal{Y}$, where $\mathcal{M}$ is the metric space and $\mathcal{Y}$ is the signal space, DPF effectively unifies the approach for handling data across different domains.

Empirical Evaluation

2D Images

The model's performance on the CelebA-HQ and CIFAR-10 datasets reveals that DPF competes favorably with existing approaches. Specifically, it surpasses other domain-agnostic methods like Functa and GEM in metrics such as FID and Precision, although it does not achieve domain-specific levels seen with models like StyleGAN2.

3D Geometry

In tasks involving 3D geometry on the ShapeNet dataset, DPF demonstrates superior performance over methods like GEM in coverage metrics, indicating its capability to model complex object distributions effectively.

Spherical Data

DPF's versatility is further highlighted in applications involving data defined on spherical surfaces, showcasing its generative capability without constraining to Euclidean geometry.

Theoretical Implications

The approach opens potential avenues for broadening the application of generative models beyond classical assumptions of data structure. By enabling modeling over metric spaces, it provides a foundation for exploring data representations that naturally align with their inherent geometric properties.

Future Directions

While DPF establishes a significant stride in generative modeling, further exploration can enhance its efficiency:

• Efficient Sampling: Adaptation of faster sampling techniques could mitigate the computational intensity typical in diffusion models.
• Architectural Optimization: Investigation into alternative architectures could further streamline processing for high-complexity data.

In conclusion, DPF provides a compelling framework that unifies generative modeling across diverse data domains, representing a noteworthy advancement in the handling of complex, continuous data structures. The model promises extensive applicability, especially in fields where data cannot be naturally expressed within Euclidean frameworks, paving the way for innovative applications in generative AI.