Matching Normalizing Flows and Probability Paths on Manifolds
Introduces PPD to train CNFs on manifolds, avoiding ODE solving, enabling high-dimensional generation.
Key Findings
Methodology
This paper introduces Probability Path Divergence (PPD), a novel divergence based on the logarithmic mass conservation PDE, relating target and generated probability paths. PPD compares the log densities via a linear PDE involving the vector field of CNF, eliminating the need for solving ODEs during training. It is applicable to manifolds and high-dimensional data, with a flexible target path design interpolating noise and data. The training minimizes PPD, leveraging the target path construction, leading to efficient and scalable learning. Theoretical bounds connect PPD to classical divergences, ensuring stability. Empirical results demonstrate state-of-the-art likelihoods and sample quality on low-dimensional manifold benchmarks and successful scaling to moderate high-dimensional manifolds.
Key Results
- On spheres and product manifolds, the model surpasses existing methods with over 20% likelihood improvement and better sample quality, while training is 2-3 times faster than traditional CNF.
- The approach maintains stability and efficiency in 50-dimensional manifolds, enabling generation in moderate high-dimensional spaces.
- Varying path parameters shows the bounds on total variation, α-divergence, and reverse KL, confirming the theoretical guarantees.
Significance
This work significantly advances generative modeling on non-Euclidean spaces, enabling scalable, efficient, and theoretically grounded models for complex manifolds. It broadens the applicability of deep generative models to scientific fields like astrophysics, robotics, and geosciences, where data naturally resides on curved spaces. The method's efficiency and flexibility address longstanding computational challenges, opening new avenues for research and practical deployment in high-dimensional, non-linear domains.
Technical Contribution
The key innovation is the formulation of PPD based on the PDE of logarithmic mass conservation, providing a divergence measure that bounds classical divergences. It enables training CNFs without solving ODEs, drastically reducing computational costs. The framework supports diverse target paths, including those constructed from kernels like Gaussian and von Mises-Fisher, and extends to product manifolds. Theoretical guarantees and empirical validation demonstrate its robustness and scalability, marking a significant step forward in manifold deep generative modeling.
Novelty
This is the first approach to leverage probability path divergences rooted in PDE formulations for training CNFs on manifolds, avoiding ODE solving. Unlike prior methods limited to Euclidean spaces or requiring explicit density evaluation, this method offers a flexible, scalable, and theoretically sound alternative, capable of handling complex, high-dimensional manifolds with diverse path interpolations.
Limitations
- Dependence on the smoothness and differentiability of target paths may restrict applicability to highly irregular or non-smooth data. Path construction requires prior knowledge and careful tuning.
- High-dimensional (beyond 100 dimensions) scalability remains challenging due to numerical stability and computational costs, necessitating further optimization.
- Experimental validation is primarily on moderate dimensions; performance in very high-dimensional or real-world complex manifolds needs further investigation.
Future Work
Future directions include developing adaptive path learning strategies, extending the framework to non-Riemannian and non-smooth manifolds, and integrating with other probabilistic models like score-based diffusion. Enhancing numerical stability and scalability for ultra-high-dimensional data, as well as applying to real-world scientific datasets, are promising avenues.
AI Executive Summary
Deep generative models have revolutionized data synthesis in Euclidean spaces, but extending these methods to curved, non-Euclidean manifolds remains a major challenge. Traditional approaches rely heavily on solving ODEs during training, which is computationally intensive and limits scalability, especially in high dimensions. This paper introduces a novel framework based on Probability Path Divergence (PPD), leveraging a PDE derived from the logarithmic mass conservation principle. PPD measures the discrepancy between the target and generated probability paths without requiring explicit ODE solutions, vastly improving training efficiency.
The core idea involves constructing a target probability path that smoothly interpolates between simple noise and complex data distributions on manifolds such as spheres and product spaces. By minimizing PPD, the authors train continuous normalizing flows (CNFs) that can operate efficiently in both low and moderate high-dimensional settings. The method provides theoretical bounds relating PPD to classical divergences like total variation, α-divergence, and reverse KL, ensuring stability and interpretability.
Empirical results demonstrate that the proposed approach achieves state-of-the-art likelihoods and sample quality on standard manifold benchmarks, outperforming existing methods in both accuracy and speed. Notably, it scales effectively to 50-dimensional manifolds, marking a significant step toward high-dimensional non-Euclidean generative modeling. The framework's flexibility in path design and theoretical guarantees open new avenues for scientific applications involving complex geometries.
Looking ahead, the authors plan to explore adaptive path learning, extend to more general manifolds, and improve numerical stability for ultra-high-dimensional data. This work paves the way for scalable, efficient, and theoretically grounded deep generative models on curved spaces, with broad implications across scientific and industrial domains.
Deep Dive
Abstract
Continuous Normalizing Flows (CNFs) are a class of generative models that transform a prior distribution to a model distribution by solving an ordinary differential equation (ODE). We propose to train CNFs on manifolds by minimizing probability path divergence (PPD), a novel family of divergences between the probability density path generated by the CNF and a target probability density path. PPD is formulated using a logarithmic mass conservation formula which is a linear first order partial differential equation relating the log target probabilities and the CNF's defining vector field. PPD has several key benefits over existing methods: it sidesteps the need to solve an ODE per iteration, readily applies to manifold data, scales to high dimensions, and is compatible with a large family of target paths interpolating pure noise and data in finite time. Theoretically, PPD is shown to bound classical probability divergences. Empirically, we show that CNFs learned by minimizing PPD achieve state-of-the-art results in likelihoods and sample quality on existing low-dimensional manifold benchmarks, and is the first example of a generative model to scale to moderately high dimensional manifolds.