Manifold Diffusion Fields (2305.15586v2)

Abstract: We present Manifold Diffusion Fields (MDF), an approach that unlocks learning of diffusion models of data in general non-Euclidean geometries. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. In addition, we show that MDF generalizes to the case where the training set contains functions on different manifolds. Empirical results on multiple datasets and manifolds including challenging scientific problems like weather prediction or molecular conformation show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

• The paper introduces Manifold Diffusion Fields, extending diffusion models to non-Euclidean spaces with spectral geometric integration.
• It develops an intrinsic coordinate system via Laplace-Beltrami eigenfunctions to robustly represent manifold data invariant to rigid transformations.
• Empirical results show MDF's superior performance in diverse applications, including climate prediction, image generation, and molecular modeling.

Manifold Diffusion Fields: Expanding the Horizon of Learning Functions on Riemannian Manifolds

The paper introduces a novel approach termed Manifold Diffusion Fields (MDF), proposing a significant extension of diffusion models for learning probability distributions of functions on non-Euclidean geometries, namely Riemannian manifolds. Previous generative models predominantly catered to Euclidean domains such as images, text, and videos, but substantial real-world scientific problems often exist in curved spaces. MDF leverages geometric insights from spectral analysis, particularly using the eigen-functions of the Laplace-Beltrami Operator (LBO), to develop an intrinsic coordinate system for these manifolds.

Core Methodological Contributions

1. Spectral Geometric Integration: MDF incorporates spectral geometry by utilizing the eigen-decomposition of the LBO to define an intrinsic and robust coordinate system for manifolds. This addresses the challenge of representing manifold data invariant to rigid transformations.
2. Generative Model Framework: MDF formulates a diffusion generative model over these manifolds, enabling the learning and sampling of function distributions over manifolds with varying geometries.
3. Empirical Validation: The authors establish the applicability of MDF through empirical evidence on datasets spanning climate prediction, MNIST digits, and CelebA-HQ images on different manifolds. Notably, MDF exhibits superior performance to traditional models in preserving the diversity and fidelity of generated fields across manifolds with various curvatures.

Empirical Results and Observations

MDF outperforms previous methods such as DPF and GASP when tasked with the generation of functions mapped over more complex geometric domains. Specifically, the manifold's mean curvature plays a significant role, where MDF maintains performance even as complexity increases. Additionally, the results on molecular conformer generation underscore MDF's capability to generalize across multiple potential manifolds, capturing the underlying distribution with high precision—a testament to its robustness and versatility.

Implications and Future Directions

The intrinsic robustness of MDF to various transformations paves the way for applications across numerous scientific domains ranging from computational chemistry to geospatial sciences. In particular, the ability to handle datasets with differing manifolds opens up possibilities for more generalized foundation models for complex data-laden tasks. This capability is especially crucial for addressing forward and inverse problems in partial differential equations (PDEs) on manifold structures—a natural fit given MDF's construction.

A potential avenue for further exploration lies in optimizing the computational demands inherent in dealing with the manifold's high-dimensional embeddings and achieving computational efficiency during the sampling processes. Moreover, extending MDF to address unsupervised or semi-supervised tasks on manifolds could provide a broader framework for manifold-based learning applications.

In conclusion, Manifold Diffusion Fields offer a cohesive framework for extending the efficacy of diffusion models beyond conventional domains into the complex geometries of Riemannian manifolds. This advancement not only broadens the scope of function learning but also provides a robust toolkit for navigating the intricacies of manifold data, serving as a potential cornerstone for future innovations in manifold-based machine learning applications.