  Constructing an efficient parameterization 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 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 samples from a
linear subspace. To process noisy data samples from a nonlinear manifold, 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 and non-asymptotic
random matrix theory, we study the stability of the subspace estimated by PCA
as a function of scale, and bound (with high probability) the angle it forms
with the true tangent space. By adaptively selecting the scale that minimizes
this bound, our analysis reveals an appropriate scale for local tangent plane
recovery. We also introduce a geometric uncertainty principle quantifying the
limits of noise-curvature perturbation for stable recovery. With the purpose of
providing perturbation bounds that can be used in practice, we propose plug-in
estimates that make it possible to directly apply the theoretical results to
real data sets.
