A New Algorithm for Enumeration of Cells of Hyperplane Arrangements and a Comparison with Avis and Fukuda's Reverse Search

Rada, Miroslav; Černý, Michal

doi:10.1137/15m1027930

Cited by 16 publications

(12 citation statements)

References 6 publications

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“…Avis and Fukuda (1996) were the first to provide an enumeration algorithm that runs in a time proportional to the maximum number of sets with non-empty interior. Improvements to this algorithm were made by Sleumer (1999) and Rada and Cerny (2018). The algorithm of Gu and Koenker (2020) is most closely related to the latter paper, and was developed for the problem of nonparametric maximum likelihood in a linear random coefficient model.…”

Section: Implementation and Hyperplane Arrangementmentioning

confidence: 99%

“…To address this issue, the algorithm proposed in Gu and Koenker (2020) builds upon the algorithm in Rada and Cerny (2018). The idea is to add one hyperplane at a time, and to enumerate all feasible response types after adding each new hyperplane.…”

Section: It Runs In a Time Proportional To O(m D θ )mentioning

confidence: 99%

“…They then introduce a new hyperplane into the arrangement of hyperplanes, and determine all newly created response types by solving a linear program. The algorithm of Rada and Cerny (2018) requires solving a linear programming problem for all of the existing cells at each iteration, which amounts to solving O(m d θ +1 ) such problems. When m is large, which is typically the case in practice, this can become costly.…”

Section: It Runs In a Time Proportional To O(m D θ )mentioning

confidence: 99%

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Partial Identification in Nonseparable Binary Response Models with Endogenous Regressors

Gu¹,

Russell²

2021

Preprint

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This paper considers (partial) identification of a variety of parameters, including counterfactual choice probabilities, in a general class of binary response models with possibly endogenous regressors. Importantly, our framework allows for nonseparable index functions with multi-dimensional latent variables, and does not require parametric distributional assumptions. We demonstrate how various functional form, independence, and monotonicity assumptions can be imposed as constraints in our optimization procedure to tighten the identified set, and we show how these assumptions have meaningful interpretations in terms of restrictions on latent types. In the special case when the index function is linear in the latent variables, we leverage results in computational geometry to provide a tractable means of constructing the sharp set of constraints for our optimization problems. Finally, we apply our method to study the effects of health insurance on the decision to seek medical treatment.

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Section: Implementation and Hyperplane Arrangementmentioning

confidence: 99%

Section: It Runs In a Time Proportional To O(m D θ )mentioning

confidence: 99%

Section: It Runs In a Time Proportional To O(m D θ )mentioning

confidence: 99%

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Partial Identification in Nonseparable Binary Response Models with Endogenous Regressors

Gu¹,

Russell²

2021

Preprint

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“…is a linear programming problem which can be solved efficiently (indeed, the constraints "β ∈ C π " are linear). The problem (7) of minimizing F (β) reduces to an enumeration of π ∈ C. What remains is to show that the cells π ∈ C can be enumerated in a reasonable way so that not all potential permutations π ∈ S n are to be tested. Observe that if p n, there are many permutations π ∈ S n such that π ∈ C; this follows from the bound (6).…”

Section: High-level Outline Of the Algorithmmentioning

confidence: 99%

A class of optimization problems motivated by rank estimators in robust regression

et al. 2020

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A rank estimator in robust regression is a minimizer of a function which depends (in addition to other factors) on the ordering of residuals but not on their values. Here we focus on the optimization aspects of rank estimators. We distinguish two classes of functions: the class with a continuous and convex objective function (CCC), which covers the class of rank estimators known from statistics, and also another class (GEN), which is far more general. We propose efficient algorithms for both classes. For GEN we propose an enumerative algorithm that works in polynomial time as long as the number of regressors is O(1). The proposed algorithm utilizes the special structure of arrangements of hyperplanes that occur in our problem and is superior to other known algorithms in this area. For the continuous and convex case, we propose an unconditionally polynomial algorithm finding the exact minimizer, unlike the heuristic or approximate methods implemented in statistical packages.

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“…Sleumer (1998) improved upon their reverse search algorithm. More recently, Rada and Černý (2018) have proposesd an incremental enumeration algorithm that is asymptotically equivalent to the Avis-Fukuda's reverse search algorithm, but is demonstrably faster in finite samples. The most costly component of the Rada-Černý algorithm involves solving the linear programs (3).…”

Section: Multivariate Randomnessmentioning

confidence: 99%

Nonparametric maximum likelihood methods for binary response models with random coefficients

Gu¹,

Koenker²

2018

Preprint

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The venerable method of maximum likelihood has found numerous recent applications in nonparametric estimation of regression and shape constrained densities. For mixture models the NPMLE of Kiefer and Wolfowitz (1956) plays a central role in recent development of empirical Bayes methods. The NPMLE has also been proposed by Cosslett (1983) as an estimation method for single index linear models for binary response with random coefficients. However, computational difficulties have hindered its application. Combining recent developments in computational geometry and convex optimization we develop a new approach to computation for such models that dramatically increases their computational tractability. Consistency of the method is established for an expanded profile likelihood formulation. The methods are evaluated in simulation experiments, compared to the deconvolution methods of Gautier and Kitamura (2013) and illustrated in an application to modal choice for journey-to-work data in the Washington DC area.

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A New Algorithm for Enumeration of Cells of Hyperplane Arrangements and a Comparison with Avis and Fukuda's Reverse Search

Cited by 16 publications

References 6 publications

Partial Identification in Nonseparable Binary Response Models with Endogenous Regressors

Partial Identification in Nonseparable Binary Response Models with Endogenous Regressors

A class of optimization problems motivated by rank estimators in robust regression

Nonparametric maximum likelihood methods for binary response models with random coefficients

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