Curvature Estimation on Data Manifolds via Diffusion-augmented Sampling

Jason Wang, Bobak Kiani, Melanie Weber

Keywords: curvature estimation, scalar curvature, normal curvature, diffusion model

TL;DR: We propose diffusion models to more accurately estimate curvature of a data manifold.

Abstract: Data geometry is fundamental to machine learning and data analysis, yet practical tools for characterizing the geometry of data manifolds remain limited. While intrinsic dimension estimation is well-studied, curvature, a key measure of local manifold structure, is far harder to approximate from noisy, sparsely sampled data. We introduce a diffusion-based framework for curvature estimation aiming to mitigate challenges due to low sample density. We train a diffusion model to learn a latent representation of the manifold, which we then probe to augment the raw dataset and obtain a denser sample. Compared to state-of-the-art curvature estimators applied directly to the raw data, diffusion-augmented methods achieve superior performance on heterogeneous manifolds when using high-fidelity diffusion models.

Submission Number: 87