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29 March 2024
 
  » arxiv » 1910.5084

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Fitting a manifold of large reach to noisy data
Charles Fefferman ; Sergei Ivanov ; Matti Lassas ; Hariharan Narayanan ;
Date 11 Oct 2019
AbstractLet ${mathcal M}subset {mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${mathcal M}$ is greater than $ au$. Moreover, let $mu$ be a probability measure on ${mathcal M}$ which density on ${mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_jin {mathcal M}$, $j=1,2,dots,N$ be $N$ independent random samples from distribution $mu$. Also, let $xi_j$, $j=1,2,dots, N$ be independent random samples from a Gaussian random variable in ${mathbb R}^n$ having covariance $sigma^2I$, where $sigma$ is less than a certain specified function of $d, V$ and $ au$. We assume that we are given the data points $y_j=x_j+xi_j,$ $j=1,2,dots,N$, modelling random points of ${mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${mathcal M}_osubset {mathbb R}^n$ whose Hausdorff distance to ${mathcal M}$ is less than $Cdsigma^2/ au$ and whose reach is greater than $c{ au}/d^6$ with universal constants $C,c > 0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on $sigma^{-1}$ and exponentially on $d$
Source arXiv, 1910.5084
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