Software for Exact Integration of Polynomials over Polyhedra

Loera, Jesús A. De; Dutra, Brandon; Koeppe, Matthias; Moreinis, S.; Pinto, Gregory; Wu, Jian-Qiu

doi:10.48550/arxiv.1108.0117

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…There are at least two software packages which have implemented Barvinok's algorithm, namely LATTE [71] and BARVINOK [69,72]. In §I we have reported on the performance of our implementation of Algorithm V.1 for computing Kronecker coefficients using the latter package.…”

Section: Computed By Barvinok's Algorithm (See Discussion Above)mentioning

confidence: 99%

Computing Multiplicities of Lie Group Representations

Christandl

Doran

Walter

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Abstract-For fixed compact connected Lie groups H ⊆ G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes.The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Nonasymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.

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Section: Computed By Barvinok's Algorithm (See Discussion Above)mentioning

confidence: 99%

Computing Multiplicities of Lie Group Representations

Christandl

Doran

Walter

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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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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“…The generalized Ehrhart function is given by a quasipolynomial q(k) of degree ≤ deg f + rank M − 1, and the coefficient of k deg f +rank M−1 in q(k) can easily be described as the integral of the highest homogeneous component of f over the polytope P. Therefore we have also included (and implemented) an approach to the computation of integrals of polynomials over rational polytopes in the spirit of the Ehrhart series computation. See [2] and [8] for more sophisticated approaches.…”

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confidence: 99%