Tangent Estimation from Point Samples

Cheng, Siu-Wing; Chiu, Man-Kwun

doi:10.1007/s00454-016-9809-z

Cited by 7 publications

(9 citation statements)

References 20 publications

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“…When k ≥ 3, the procedure (2) outperforms PCA-based estimators of [31] and [33], where convergence rates of the form (1/n) β with 1/d < β < 2/d are obtained. This bound also recovers the result of [11] in the case k = 3, where a similar procedure is used. When the noise level σ is of order h α , with 1 ≤ α ≤ k, Theorem 2 yields a convergence rate in h α−1 .…”

Section: Tangent Spacessupporting

confidence: 79%

“…, T k,j ) in this basis under orthogonality constraints. It is worth mentioning that a similar problem is explicitly solved in [11], leading to an optimal tangent space estimation procedure in the case k = 3.…”

Section: Resultsmentioning

confidence: 99%

“…When M is smoother, it has been proved in [11] that a convergence rate in n −2/d might be achieved, based on the existence of a local order 3 Taylor expansion of the submanifold on top of its tangent spaces. Thus, a natural extension of the C 2 τ min model to C k -submanifolds should ensure that such an expansion exists at order k and satisfies some regularity constraints.…”

Section: Reach and Regularity Of Submanifoldsmentioning

confidence: 99%

“…This is pretty clear for tangent space estimation, where convergence rates of PCA-based estimators range from (1/n) 1/d in the C 2 case [1] to (1/n) α with 1/d < α < 2/d in more regular settings [31,33]. In addition, it seems that PCA-based estimators are outperformed by estimators taking into account higher orders of smoothness [11,9], for regularities at least C 3 . For instance fitting quadratic terms leads to a convergence rate of order (1/n) 2/d in [11].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Nonasymptotic rates for manifold, tangent space and curvature estimation

Aamari¹,

Levrard²

2019

Ann. Statist.

166

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Given a noisy sample from a submanifold M ⊂ R D , we derive optimal rates for the estimation of tangent spaces TX M , the second fundamental form II M X , and the submanifold M . After motivating their study, we introduce a quantitative class of C ksubmanifolds in analogy with Hölder classes. The proposed estimators are based on local polynomials and allow to deal simultaneously with the three problems at stake. Minimax lower bounds are derived using a conditional version of Assouad's lemma when the base point X is random.

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Section: Tangent Spacessupporting

confidence: 79%

Section: Resultsmentioning

confidence: 99%

Section: Reach and Regularity Of Submanifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonasymptotic rates for manifold, tangent space and curvature estimation

Aamari¹,

Levrard²

2019

Ann. Statist.

166

View full text Add to dashboard Cite

show abstract

“…For an i.i.d. point cloud X n , asymptotic and nonasymptotic rates of tangent space estimation derived in C 3 -like models can be found in [2,14,36], yielding bounds on sin θ of order (log n/n) 1/d . In that case, the typical scale of minimum interpoint distance is δ n −2/d , as stated in the asymptotic result Theorem 2.1 in [29] for the flat case of R d .…”

Section: Towards Unknown Tangent Spacesmentioning

confidence: 99%

Estimating the reach of a manifold

Aamari¹,

Kim²,

Chazal³

et al. 2019

Electron. J. Statist.

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Various problems in manifold estimation make use of a quantity called the reach, denoted by τ M , which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimatorτ of τ M is proposed in an oracle framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud Xn, τ (Xn) is showed to achieve uniform expected loss bounds over a C 3 -like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach.MSC 2010 subject classifications: Primary 62G05; secondary 62C20, 68U05.

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Implicit Manifold Reconstruction

Cheng

Chiu

2019

Discrete Comput Geom

Self Cite

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Let M ⊂ R d be a compact, smooth and boundaryless manifold with dimension m and unit reach. We show how to construct a function ϕ : R d → R d−m from a uniform (ε, κ)sample P of M that offers several guarantees. Let Z ϕ denote the zero set of ϕ. Let M denote the set of points at distance ε or less from M. There exists ε 0 ∈ (0, 1) that decreases as d increases such that if ε ≤ ε 0 , the following guarantees hold. First, Z ϕ ∩ M is a faithful approximation of M in the sense that Z ϕ ∩ M is homeomorphic to M, the Hausdorff distance between Z ϕ ∩ M and M is O(m 5/2 ε 2 ), and the normal spaces at nearby points in Z ϕ ∩ M and M make an angle O(m 2 √ κε). Second, ϕ has local support; in particular, the value of ϕ at a point is affected only by sample points in P that lie within a distance of O(mε). Third, we give a projection operator that only uses sample points in P at distance O(mε) from the initial point. The projection operator maps any initial point near P onto Z ϕ ∩ M in the limit by repeated applications. * A preliminary version appears in where e g / R · ∆x converges to the zero vector as R · ∆x → 0. Since R is fixed, it means that e g / ∆x tends to the zero vector as ∆x → 0. We multiply both sides of (27) by R t and then subtract the resulting equation from (26). Some terms cancel each other because f (x + ∆x) = R t · g (R · (x + ∆x)) and f (x) = R t · g(x ) = R t · g (R · x). We obtainTherefore,We are free to choose the direction of ∆x. We choose it such that· R · ∆x , i.e., ∆x is an eigenvector corresponding to the largest eigenvalue of J f (x) − R t · J g (x ) · R. Then,

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Tangent Estimation from Point Samples

Cited by 7 publications

References 20 publications

Nonasymptotic rates for manifold, tangent space and curvature estimation

Nonasymptotic rates for manifold, tangent space and curvature estimation

Estimating the reach of a manifold

Implicit Manifold Reconstruction

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