Fast Local Mean Curvature Estimation for High-Dimensional Data Manifolds

Local mean curvature is a powerful geometric descriptor of data manifolds. Nonetheless, its computational cost has largely prevented its adoption in modern machine learning methods. In the present work, it is introduced a scalable curvature estimation framework for high-dimensional data (MeCuCo). The new method combines an exact algebraic reformulation of a classical curvature estimator with a low-rank approximation that exploits the structure of local neighborhoods. This innovative combination yields a reduced computational complexity – from quartic to nearly linear in the ambient dimension for typical settings, while preserving the quality of the estimated curvature values. Theoretical analysis of the proposed formulation and approximation are provided, resulting in outstanding speed-up improvements between 50 and 300 times on real-world datasets. Importantly, through rendering curvature estimation computationally tractable at scale, MeCuCo facilitates the integration of curvature-aware representations and geometric inductive biases into manifold learning, representation learning, and related machine learning paradigms.