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

UR - http://www.scopus.com/inward/citedby.url?scp=85167975007&partnerID=8YFLogxK

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 -