Exorcising the Curse of Dimensionality for Curvature Estimation

Speaker: Jiayi Chen (Dartmouth)

Date: 5/20/25

Abstract: In manifold learning, data are often modeled as noisy samples from an underlying smooth manifold. While local linear approximations capture some geometry, estimating curvature—a finer and nonlinear feature—remains challenging, especially in high dimensions. We show that naive curvature estimates suffer from a strong bias that grows rapidly with dimension, a manifestation of the “curse of dimensionality.” Focusing on curvature defined via variation in estimated tangent spaces, we provide a probabilistic analysis explaining this bias and introduce a corrected estimator using a push-forward of the noise model. Experiments on high-dimensional spheres demonstrate substantial accuracy gains. Our method suggests a potential path toward mitigating bias in discrete curvature estimation.