Fitting a manifold of large reach to noisy data

Abstract

Let ℳ ⊂ ℝn be a C2-smooth compact submanifold of dimension d. Assume that the volume of ℳ is at most V and the reach (i.e. the normal injectivity radius) of ℳ is greater than τ. Moreover, let μ be a probability measure on ℳ whose density on ℳ is a strictly positive Lipschitz-smooth function. Let xj ∈ ℳ j = 1, 2,⋯,N be N independent random samples from distribution μ. Also, let ζj, j = 1, 2,⋯,N be independent random samples from a Gaussian random variable in ℝn having covariance σ2I, where σ is less than a certain specified function of d,V and τ. We assume that we are given the data points yj = xj + ζj, j = 1, 2,⋯,N, modeling random points of ℳ with measurement noise. We develop an algorithm which produces from these data, with high probability, a d dimensional submanifold ℳo ⊂ ℝn whose Hausdorff distance to ℳ is less than Δ for Δ > Cdσ2/τ and whose reach is greater than cτ/d6 with universal constants C,c > 0. The number N of random samples required depends almost linearly on n, polynomially on Δ-1 and exponentially on d.