toplogo
Sign In

Efficient Simulation of Diffusion Bridge Processes on Sub-Riemannian Manifolds using Score Matching


Core Concepts
This paper presents a method for efficiently simulating diffusion bridge processes on sub-Riemannian manifolds by leveraging recent progress in machine learning to train neural network approximations of the score function.
Abstract
The paper addresses the problem of simulating diffusion bridge processes on sub-Riemannian manifolds, which is challenging due to the hypoellipticity of the underlying diffusion processes. Key highlights: Diffusion bridge processes are essential for inference, data imputation, and geometric statistics, but existing methods do not directly apply to sub-Riemannian manifolds. The authors demonstrate how recent score matching techniques can be generalized to the sub-Riemannian setting to train neural network approximations of the score function. This allows for the simulation of conditioned diffusion processes on sub-Riemannian manifolds by using the learned score in the stochastic differential equation for the bridge process. The authors derive a sub-Riemannian denoising loss function that accounts for the non-holonomic nature of the horizontal distribution, which is a key difference compared to the Euclidean case. Numerical experiments on the Heisenberg group showcase the effectiveness of the proposed approach for sampling from the bridge process and demonstrating its concentration in small time.
Stats
The paper does not contain any explicit numerical data or statistics. It focuses on the theoretical development of the method and provides numerical examples to illustrate the approach.
Quotes
"Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics." "As far as the authors are aware, it has not been known how to effectively simulate bridge processes on sub-Riemannian manifolds." "We rely on Taylor expansion for the stochastic integral to approximate this component, which we then exploit to derive a sub-Riemannian denoising loss for use in the training of neural network score approximators."

Key Insights Distilled From

by Erlend Grong... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.15258.pdf
Score matching for sub-Riemannian bridge sampling

Deeper Inquiries

How can the proposed score matching approach be extended to other types of geometric structures beyond sub-Riemannian manifolds, such as Finsler or Carnot-Carathéodory spaces

The proposed score matching approach can be extended to other types of geometric structures beyond sub-Riemannian manifolds, such as Finsler or Carnot-Carathéodory spaces, by adapting the methodology to the specific characteristics of these spaces. In Finsler geometry, for example, where the metric tensor varies with direction, the score approximation would need to account for this non-linearity in the metric. Similarly, in Carnot-Carathéodory spaces, which are sub-Riemannian manifolds with additional restrictions on the horizontal distribution, the score matching approach would need to consider these constraints in the approximation process. By adjusting the neural network architecture and loss functions to reflect the unique properties of these geometric structures, the score matching technique can be effectively applied to a broader range of spaces.

What are the potential applications of efficiently simulating diffusion bridge processes on sub-Riemannian manifolds in fields like robotics, control theory, or quantum physics

Efficiently simulating diffusion bridge processes on sub-Riemannian manifolds has numerous potential applications in various fields. In robotics, the ability to simulate conditioned diffusion processes is crucial for motion planning and control of robotic systems operating in constrained environments, such as robot arms or autonomous vehicles navigating complex terrains. In control theory, simulating diffusion bridge processes can aid in designing optimal control strategies for systems with stochastic dynamics, allowing for more robust and adaptive control algorithms. In quantum physics, where stochastic processes play a significant role in modeling quantum systems, simulating diffusion bridges can provide insights into the behavior of quantum particles and the evolution of quantum states over time. Overall, the applications span a wide range of disciplines where understanding and modeling stochastic processes are essential.

Can the insights from this work on sub-Riemannian geometry be leveraged to improve the performance of diffusion-based generative models on manifolds in machine learning

The insights from this work on sub-Riemannian geometry can indeed be leveraged to improve the performance of diffusion-based generative models on manifolds in machine learning. By incorporating the knowledge of sub-Riemannian structures and the challenges they pose into the design of generative models, researchers can develop more accurate and efficient models for generating data on complex geometric spaces. The understanding of hypoellipticity, horizontal distributions, and geometric constraints in sub-Riemannian manifolds can inform the architecture of neural networks used in generative modeling, leading to better representations of data and improved sampling techniques. Additionally, the techniques developed for simulating diffusion bridge processes can be adapted to enhance the training and performance of generative models on manifolds, enabling more effective data generation and representation learning in machine learning applications.
0