Curvature-Aware Regularization on Riemannian Submanifolds

One fundamental assumption in object recognition as well as in other computer vision and pattern recognition problems is that the data generation process lies on a manifold and that it respects the intrinsic geometry of the manifold. This assumption is held in several successful algorithms for diffusion and regularization, in particular, in graph-Laplacian-based algorithms. We claim that the performance of existing algorithms can be improved if we additionally account for how the manifold is embedded within the ambient space, i.e., if we consider the extrinsic geometry of the manifold. We present a procedure for characterizing the extrinsic (as well as intrinsic) curvature of a manifold M which is described by a sampled point cloud in a high-dimensional Euclidean space. Once estimated, we use this characterization in general diffusion and regularization on M, and form a new regularizer on a point cloud. The resulting re-weighted graph Laplacian demonstrates superior performance over classical graph Laplacian in semi-supervised learning and spectral clustering.