Optimal Tangent Plane Recovery From Noisy Manifold Samples

Details Optimal Tangent Plane Recovery From Noisy Manifold Samples Optimal Tangent Plane Recovery From Noisy Manifold Samples

2011-11-20

arxiv.org/abs/1111.4601v1

Physics

Data Analysis, Statistics and Probability

Submitted

Scientific paper

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 lane. 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 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 the optimal scale for local tangent plane recovery.

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