Fitting a C<sup>m</sup>-smooth function to data, I

Fefferman, Charles

doi:10.4007/annals.2009.169.315

Cited by 44 publications

(5 citation statements)

References 24 publications

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“…We refer to the introductions of the papers [3,5,6,24,38] for some motivation and background for this problem. We also recommend to see [8,9,11,15,16,17,18,19,20,21,22,23,26,29,28,30,32,33,34,35,36,37,39,40] and the references therein for information about general (we mean not necessarily convex) Whitney extension problems for jets and for functions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Locally $C^{1,1}$ convex extensions of $1$-jets

Azagra¹

2019

Preprint

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Let E be an arbitrary subset of R n , and f : E → R, G : E → R n be given functions. We provide necessary and sufficient conditions for the existence of a convex function F ∈ C 1,1 loc (R n ) such that F = f and ∇F = G on E. We give a useful explicit formula for such an extension F , and a variant of our main result for the class C 1,ω loc , where ω is a modulus of continuity. We also present two applications of these results, concerning how to find C 1,1 loc convex hypersurfaces with prescribed tangent hyperplanes on a given subset of R n , and some explicit formulas for (not necessarily convex) C 1,1 loc extensions of 1-jets.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Locally $C^{1,1}$ convex extensions of $1$-jets

Azagra¹

2019

Preprint

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“…Theorems 1 and 2 are progress toward the proof of a Finiteness Principle for the non-linear space of smooth convex function C 1,1 c (R n ). Our hope is this work and the continued study of finiteness principles for smooth convex functions allow the development of algorithms for constructing smooth, convex extensions of a function (or its approximation) analogous to the work by C. Fefferman and Boaz Klartag in [9,10] for…”

Section: G(x)−g(y)| |X−y|mentioning

confidence: 96%

Finiteness Principles for Smooth Selection

2016

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Let E ⊂ R n be a compact set, and f : E → R. How can we tell if there exists a convex extensionAssuming such an extension exists, how small can one take the Lipschitz constant Lip(∇F ) := sup x,y∈R n ,x =yWe provide an answer to these questions for the class of strongly convex functions by proving that there exist constants k # ∈ N and C > 0 depending only on the dimension n, such that if for every subset S ⊂ E, #S ≤ k # , there exists an η-strongly convex functionand Lip(∇F ) ≤ CM 2 /η. Further, we prove a Finiteness Principle for the space of convex functions in C 1,1 (R) and that the sharp finiteness constant for this space is k # = 5.

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“…The solution of Whitney's problems has led to a new algorithm for interpolation of data, due to C. Fefferman and B. Klartag [40,41], where the authors show how to compute efficiently an interpolant F (x), whose C m norm lies within a factor C of least possible, where C is a constant depending only on m and n.…”

Section: A Survey Of Related Workmentioning

confidence: 99%

Reconstruction of a Riemannian Manifold from Noisy Intrinsic Distances

Fefferman¹,

Ivanov²,

Lassas³

et al. 2020

SIAM Journal on Mathematics of Data Science

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We consider the reconstruction of a manifold (or, invariant manifold learning), where a smooth Riemannian manifold M is determined from the intrinsic distances (that is, geodesic distances) of points in a discrete subset of M . In the studied problem, the Riemannian manifold (M, g) is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let \{ X1, X2, . . . , XN \} be a set of N sample points sampled randomly from an unknown Riemannian M manifold. We assume that we are given the numbers D jk = dM (Xj, X k ) + \eta jk , where j, k \in \{ 1, 2, . . . , N \} . Here, dM (Xj, X k ) are geodesic distances, and \eta jk are independent, identically distributed random variables such that the exponential moment \BbbE e | \eta jk | is finite. We show that when N \sim C0\delta - 3n (log(1/\delta )) 5 log(1/\theta ), with the probability 1 -\theta , it is possible to construct a manifold that approximates the Riemannian manifold (M, g) with the error \delta . Here, C0 depends on the intrinsic dimension n of M and the bounds for the diameter, sectional curvature, the injectivity radius of (M, g), and the the exponential moment of the noise. This problem is a generalization of the geometric Whitney problem with random measurement errors. We also consider the case when the information on the noisy distance D jk of points Xj and X k is missing with a certain probability. In particular, we consider the case when we have no information on points that are far away.

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Fitting a C^m-smooth function to data, I

Cited by 44 publications

References 24 publications

Locally $C^{1,1}$ convex extensions of $1$-jets

Locally $C^{1,1}$ convex extensions of $1$-jets

Finiteness Principles for Smooth Selection

Reconstruction of a Riemannian Manifold from Noisy Intrinsic Distances

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Fitting a Cm-smooth function to data, I

Cited by 44 publications

References 24 publications

Locally $C^{1,1}$ convex extensions of $1$-jets

Locally $C^{1,1}$ convex extensions of $1$-jets

Finiteness Principles for Smooth Selection

Reconstruction of a Riemannian Manifold from Noisy Intrinsic Distances

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Fitting a C^m-smooth function to data, I