Software for exact integration of polynomials over polyhedra

Loera, Jesús A. De; Dutra, Brandon; Köppe, Matthias; Moreinis, S.; Pinto, Gregory; Wu, Jianbing

doi:10.1145/2110170.2110175

Cited by 23 publications

(19 citation statements)

References 36 publications

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“…LRS is not useful for H-polytopes as stated on its webpage: "If the volume option is applied to an H-representation, the results are not predictable." Latte implements the same decomposition methods as VINCI; it is less prone to round-off error but slower [14]. Normaliz applies triangulation: it handles cubes for d ≤ 10, in < 1 min, but for d = 15, it did not terminate after 5 hours.…”

Section: Methodsmentioning

confidence: 99%

Efficient Random-Walk Methods for Approximating Polytope Volume

Emiris

Fisikopoulos

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytope's volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problem's geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B 11 , . . . , B 15 , whereas exact methods have only computed that of B 10 .

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Section: Methodsmentioning

confidence: 99%

Efficient Random-Walk Methods for Approximating Polytope Volume

Emiris

Fisikopoulos

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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“…Our work is related to probabilistic program analysis [13], probabilistic abstract interpretation [20], probabilistic model checking [15] and volume computations [8]. We discuss here some of the most closely related work.…”

Section: Related Workmentioning

confidence: 99%

“…[9,11,28]. Geldenhuys et al [11] considered uniform distributions for the inputs, linear integer arithmetic constraints, and used LattE Machiato [8] to count solutions of path conditions produced during symbolic execution. Sankaranarayanan et al [28] and Filieri et al [9] proposed similar techniques to compute probabilities of violating program assertions.…”

Section: Related Workmentioning

confidence: 99%

Iterative distribution-aware sampling for probabilistic symbolic execution

Borges

Filieri

d’Amorim

et al. 2015

Proceedings of the 2015 10th Joint Meeting on Foundations of Software Engineering

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Probabilistic symbolic execution aims at quantifying the probability of reaching program events of interest assuming that program inputs follow given probabilistic distributions. The technique collects constraints on the inputs that lead to the target events and analyzes them to quantify how likely it is for an input to satisfy the constraints. Current techniques either handle only linear constraints or only support continuous distributions using a "discretization" of the input domain, leading to imprecise and costly results.We propose an iterative distribution-aware sampling approach to support probabilistic symbolic execution for arbitrarily complex mathematical constraints and continuous input distributions. We follow a compositional approach, where the symbolic constraints are decomposed into sub-problems whose solution can be solved independently. At each iteration the convergence rate of the computation is increased by automatically refocusing the analysis on estimating the sub-problems that mostly affect the accuracy of the results, as guided by three different ranking strategies.Experiments on publicly available benchmarks show that the proposed technique improves on previous approaches in terms of scalability and accuracy of the results.

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“…The regime neatly separates the model enumeration from the integration, which is demonstrated by allowing a choice of two integration schemes. The first is a provably efficient and exact integration approach for polynomial densities [21,4,20] and the second is an unmodified integration library available in the programming language platform (Python in our case). The results obtained are very promising with regards to the empirical behavior: we perform competitively to the ex-isting state-of-the-art WMI solver [29].…”

Section: Introductionmentioning

confidence: 99%

Scaling up Probabilistic Inference in Linear and Non-linear Hybrid Domains by Leveraging Knowledge Compilation

Fuxjaeger

Belle

2020

Proceedings of the 12th International Conference on Agents and Artificial Intelligence

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Weighted model integration (WMI) extends weighted model counting (WMC) in providing a computational abstraction for probabilistic inference in mixed discrete-continuous domains. WMC has emerged as an assembly language for state-of-the-art reasoning in Bayesian networks, factor graphs, probabilistic programs and probabilistic databases.In this regard, WMI shows immense promise to be much more widely applicable, especially as many real-world applications involve attribute and feature spaces that are continuous and mixed. Nonetheless, state-of-the-art tools for WMI are limited and less mature than their propositional counterparts.In this work, we propose a new implementation regime that leverages propositional knowledge compilation for scaling up inference. In particular, we use sentential decision diagrams, a tractable representation of Boolean functions, as the underlying model counting and model enumeration scheme. Our regime performs competitively to state-of-theart WMI systems, but is also shown, for the first time, to handle non-linear constraints over non-linear potentials.

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Software for exact integration of polynomials over polyhedra

Cited by 23 publications

References 36 publications

Efficient Random-Walk Methods for Approximating Polytope Volume

Efficient Random-Walk Methods for Approximating Polytope Volume

Iterative distribution-aware sampling for probabilistic symbolic execution

Scaling up Probabilistic Inference in Linear and Non-linear Hybrid Domains by Leveraging Knowledge Compilation

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