Lipschitz Unimodal and Isotonic Regression on Paths and Trees

Agarwal, Pankaj K.; Phillips, Jeff M.; Sadri, Bardia

doi:10.1007/978-3-642-12200-2_34

Cited by 3 publications

(2 citation statements)

References 32 publications

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“…Recent applications of isotonic regression on trees include taxonomies and analyzing web and GIS data [1,9,22,34]. Here the isotonic ordering is towards the root.…”

Section: Treesmentioning

confidence: 99%

“…Note that when V is points on the real line, f is not necessarily defined at points not in V . For example, if all weights are the same and the (x, y) values are (1,7), (2,5), Figure 1: Isotonic Regression on 4×3 Grid (3,8), then for all p an optimal regression is 6 on [1,2] and 8 at 3, but for any x ∈ (2, 3) and y ∈ [6,8] there is an optimal isotonic regression f for which f (x) = y. Further, while isotonic regression is unique when 1 < p < ∞, for p = ∞ it is not necessarily unique.…”

Section: Introductionmentioning

confidence: 99%

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L infinity Isotonic Regression for Linear, Multidimensional, and Tree Orders

Stout

2015

Preprint

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Algorithms are given for determining L ∞ isotonic regression of weighted data. For a linear order, grid in multidimensional space, or tree, of n vertices, optimal algorithms are given, taking Θ(n) time. These improve upon previous algorithms by a factor of Ω(log n). For vertices at arbitrary positions in d-dimensional space a Θ(n log d−1 n) algorithm employs iterative sorting to yield the functionality of a multidimensional structure while using only Θ(n) space. The algorithms utilize a new non-constructive feasibility test on a rendezvous graph, with bounded error envelopes at each vertex.

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“…Recent applications of isotonic regression on trees include taxonomies and analyzing web and GIS data [1,9,22,34]. Here the isotonic ordering is towards the root.…”

Section: Treesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

L infinity Isotonic Regression for Linear, Multidimensional, and Tree Orders

Stout

2015

Preprint

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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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