Non-asymptotic analysis of tangent space perturbation

Constructing an efficient parameterization of a large, noisy data set of points lying close to asmooth manifold in high dimension remains a fundamental problem. One approach consists inrecovering a local parameterization 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 samplesfrom a linear subspace. To process noisy data samples from a non-linear manifold, PCA must beapplied locally, at a scale small enough such that the manifold is approximately linear, but at ascale large enough such that structure may be discerned from noise. Using eigenspace perturbationtheory and non-asymptotic random matrix theory, we study the stability of the subspace estimated byPCA as a function of scale, and bound (with high probability) the angle it forms with the truetangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals anappropriate scale for local tangent plane recovery. We also introduce a geometric uncertaintyprinciple quantifying the limits of noise–curvature perturbation for stable recovery. With thepurpose of providing perturbation bounds that can be used in practice, we propose plug-in estimatesthat make it possible to directly apply the theoretical results to real data sets.