Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension

TL;DR

This paper presents new diffusion geometry-based estimators for curvature, tangent spaces, and dimension of data manifolds, which are robust to noise and data sparsity, outperforming existing methods especially under noisy conditions.

Contribution

It introduces novel, noise-robust estimators for manifold curvature, tangent spaces, and dimension using diffusion geometry, with improved accuracy and no parameter tuning.

Findings

Dimension estimation outperforms existing methods on noisy benchmarks.

Tangent space and curvature estimates are parameter-free and more accurate.

Methods are robust to noise and data sparsity, outperforming classical approaches.

Abstract

We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data analysis and machine learning, it has historically been hard to implement these methods in a way that performs well statistically. Diffusion geometry lets us develop Riemannian geometry methods that are accurate and, crucially, also extremely robust to noise and low-density data. The methods we introduce here are comparable to the existing state-of-the-art on ideal dense, noise-free data, but significantly outperform them in the presence of noise or sparsity. In particular, our dimension estimate improves on the existing methods on a challenging benchmark test when even a small amount of noise is added. Our tangent space and scalar curvature estimates do not require parameter selection and substantially improve on existing techniques.