Fitting a manifold of large reach to noisy data
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Let M⊂R^n be a C^2-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 which density on M is a strictly positive Lipschitz-smooth function. Let x_j∈ M, j=1,2,...,N be N independent random samples from distribution μ. Also, let ξ_j, j=1,2,..., N be independent random samples from a Gaussian random variable in R^n having covariance σ^2I, where σ is less than a certain specified function of d, V and τ. We assume that we are given the data points y_j=x_j+ξ_j,j=1,2,...,N, modelling random points of M with measurement noise. We develop an algorithm which produces from these data, with high probability, a d dimensional submanifold M_o⊂R^n whose Hausdorff distance to M is less than Cdσ^2/τ and whose reach is greater than cτ/d^6 with universal constants C,c > 0. The number N of random samples required depends almost linearly on n, polynomially on σ^-1 and exponentially on d
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