Diffusion Processes on Implicit Manifolds

2026-04-08

AI Summary

This study addresses the problem of constructing intrinsic diffusion processes on unknown low-dimensional manifolds using only point cloud data, without access to explicit geometric information such as coordinate charts or projections. To this end, the authors propose the Implicit Manifold Diffusion (IMD) framework, which estimates the infinitesimal generator and carré-du-champ operator via a neighborhood graph to formulate a data-driven stochastic differential equation in the ambient space whose dynamics faithfully encode the intrinsic geometry of the underlying manifold. Theoretical analysis establishes that the induced probability paths weakly converge to the ideal diffusion process on the true manifold as the sample size grows. Numerically, the method is efficiently implemented using Euler–Maruyama integration. This work presents the first provably convergent manifold diffusion model in a fully implicit setting, offering both theoretical guarantees and practical tools for manifold-aware generation and sampling.

Abstract

High-dimensional data are often modeled as lying near a low-dimensional manifold. We study how to construct diffusion processes on this data manifold in the implicit setting. That is, using only point cloud samples and without access to charts, projections, or other geometric primitives. Our main contribution is a data-driven SDE that captures intrinsic diffusion on the underlying manifold while being defined in ambient space. The construction relies on estimating the diffusion's infinitesimal generator and its carré-du-champ (CDC) from a proximity graph built from the data. The generator and CDC together encode the local stochastic and geometric structure of the intended diffusion. We show that, as the number of samples grows, the induced process converges in law on the space of probability paths to its smooth manifold counterpart. We call this construction Implicit Manifold-valued Diffusions (IMDs), and furthermore present a numerical simulation procedure using Euler-Maruyama integration. This gives a rigorous basis for practical implementations of diffusion dynamics on data manifolds, and opens new directions for manifold-aware sampling, exploration, and generative modeling.

Problem

• diffusion processes
• implicit manifolds
• point cloud data
• stochastic differential equations
• manifold learning

Innovation

• Implicit Manifold
• Diffusion Process
• Stochastic Differential Equation
• Carré-du-Champ
• Manifold Learning