Smooth Interpolation of Data by Efficient Algorithms

Fefferman, Charles

doi:10.1007/978-0-8176-8376-4_4

Cited by 7 publications

(6 citation statements)

References 16 publications

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“…See for example Proposition 2 of [4]. Using the particular construction in this proof, we can take C = 6.…”

Section: Additionally Setmentioning

confidence: 99%

Algorithms for computing the optimal Lipschitz constant of interpolants with Lipschitz derivative

Hirn

2013

Preprint

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One classical measure of the quality of an interpolating function is its Lipschitz constant. In this paper we consider interpolants with additional smoothness requirements, in particular that their derivatives be Lipschitz. We show that such a measure of quality can be easily computed, giving two algorithms, one optimal in the dimension of the data, the other optimal in the number of points to be interpolated.

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“…See for example Proposition 2 of [4]. Using the particular construction in this proof, we can take C = 6.…”

Section: Additionally Setmentioning

confidence: 99%

Algorithms for computing the optimal Lipschitz constant of interpolants with Lipschitz derivative

Hirn

2013

Preprint

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“…The papers [62,63,64,65] and the references cited therein detail a complete solution to Problem (W). Observe that X ⊂ R D is arbitrary.…”

Section: Solution Problem (W)mentioning

confidence: 99%

On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment

Damelin¹

2021

Preprint

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This paper is an exposition of work of the author et al. detailing fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic-differential geometry, analysis on manifolds, complex-harmonic analysis, data science, partial differential equations, optimization and probability. A significant portion of the work is based on joint research with Charles Fefferman in the papers [39,40,41,42]. The topics of this work include (a) The space of maps of bounded mean oscillation (BMO) in R D , D ≥ 2. (b) The labeled and unlabeled near alignment and Procrustes problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (c) The Whitney near isometry extension problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (d) Partitions and clustering of compact sets and point sets with certain geometries in R D , D ≥ 2 and analysis on certain manifolds in R D , D ≥ 2. Many open problems for future research are given.

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“…[24] describes (multidimensional) L 2 convex regression algorithms based quadratic programming. Fefferman [8] studied a closely related problem of smooth interpolation of data in Euclidean space minimizing a certain norm defined on the derivatives of the function. His setup is much more general, but his algorithm cannot find arbitrarily good interpolations (ε is fixed for the algorithm).…”

Section: Introductionmentioning

confidence: 99%

Efficient Second-Order Shape-Constrained Function Fitting

Durfee

Gao

Rao

et al. 2019

Lecture Notes in Computer Science

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We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted-L ∞ norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first-and/or second-order differences. Our algorithm computes an approximation with additive error ε in O n log U ε time, where U captures the range of input values. We also give a simple greedy algorithm that runs in O(n) time for the special case of unweighted L ∞ convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming.Efficient Second-Order Shape-Constrained Function Fitting first and second derivatives of f ; their discretized equivalents are hence amenable to our new method. Shape restrictions that we cannot directly handle are studied in [28] (f is piecewise constant and the number of breakpoints is to be minimized) and [26] (unimodal f ). For a more comprehensive survey of shape-constrained function-fitting problems and their applications, see [14, §1]. Motivated by these applications, the problems have been studied in statistics (as a form of nonparametric regression), investigating, e.g., their consistency as estimators and their rate of convergence [13,14, 4].While fast algorithms for isotonic-regression variants have been designed [27], both [22] and [3] list shape constraints beyond monotonicity as important challenges. For example, fitting (multidimensional) convex functions is mostly done via quadratic or linear programming solvers [24]. In his PhD thesis, Balzs writes that current "methods are computationally too expensive for practical use, [so] their analysis is used for the design of a heuristic training algorithm which is empirically evaluated" [4, p. 1].This lack of efficient algorithms motivated the present work. Despite a few limitations discussed below (implying that we do not yet solve Balzs' problem), we give the first near-lineartime algorithms for any function-fitting problem with second-order shape constraints (such as convexity). We use dynamic programming (DP) with a novel geometric encoding for the "states". Simpler versions of such geometric DP variants were used for isotonic regression [25] and are well-known in the competitive programming community; incorporating second-order constraints efficiently is our main innovation.

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Smooth Interpolation of Data by Efficient Algorithms

Cited by 7 publications

References 16 publications

Algorithms for computing the optimal Lipschitz constant of interpolants with Lipschitz derivative

Algorithms for computing the optimal Lipschitz constant of interpolants with Lipschitz derivative

On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment

Efficient Second-Order Shape-Constrained Function Fitting

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