TY - JOUR
T1 - Fitting a manifold of large reach to noisy data
AU - Fefferman, Charles
AU - Ivanov, Sergei
AU - Lassas, Matti
AU - Narayanan, Hariharan
N1 - Publisher Copyright:
© 2023 World Scientific Publishing Company.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - Manifolds
KW - statistics
UR - http://www.scopus.com/inward/record.url?scp=85167975007&partnerID=8YFLogxK
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U2 - 10.1142/S1793525323500012
DO - 10.1142/S1793525323500012
M3 - Article
AN - SCOPUS:85167975007
SN - 1793-5253
JO - Journal of Topology and Analysis
JF - Journal of Topology and Analysis
ER -