Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension

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.