Overcoming noise, avoiding curvature: Optimal scale selection for tangent plane recovery

Constructing an efficient parametrization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parametrization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise. Using eigenspace perturbation theory, we adaptively select the scale that minimizes the angle between the subspace estimated by PCA and the true tangent space, revealing the optimal scale for local tangent plane recovery.