Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases

Z. Huang, Matthew England, J. H. Davenport, L. C. Paulson

Research output: Chapter in Book/Report/Conference proceedingConference proceeding

9 Citations (Scopus)
15 Downloads (Pure)

Abstract

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.
Original languageEnglish
Title of host publication18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
Place of PublicationCalifornia
PublisherIEEE Computer Society
Pages45-52
ISBN (Print) 978-1-5090-5707-8
DOIs
Publication statusPublished - 26 Jan 2017
EventInternational Symposium on Symbolic and Numeric Algorithms for Scientific Computing - Timisoara, Romania
Duration: 24 Sep 201627 Sep 2016

Conference

ConferenceInternational Symposium on Symbolic and Numeric Algorithms for Scientific Computing
CountryRomania
CityTimisoara
Period24/09/1627/09/16

Fingerprint

Learning systems
Decomposition
Computational geometry
Support vector machines
Experiments
Polynomials

Bibliographical note

© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. ISSN 2470-881X

Keywords

  • preconditioning
  • machine learning
  • support vector machine
  • computer algebra
  • cylindrical algebraic decomposition
  • groebner bases
  • Computers
  • Algebra
  • Machine learning algorithms
  • Support vector machines
  • Measurement
  • Electronic mail
  • Complexity theory

Cite this

Huang, Z., England, M., Davenport, J. H., & Paulson, L. C. (2017). Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases. In 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (pp. 45-52). California: IEEE Computer Society. https://doi.org/10.1109/SYNASC.2016.020

Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases. / Huang, Z.; England, Matthew; Davenport, J. H.; Paulson, L. C.

18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. California : IEEE Computer Society, 2017. p. 45-52.

Research output: Chapter in Book/Report/Conference proceedingConference proceeding

Huang, Z, England, M, Davenport, JH & Paulson, LC 2017, Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases. in 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. IEEE Computer Society, California, pp. 45-52, International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Timisoara, Romania, 24/09/16. https://doi.org/10.1109/SYNASC.2016.020
Huang Z, England M, Davenport JH, Paulson LC. Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases. In 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. California: IEEE Computer Society. 2017. p. 45-52 https://doi.org/10.1109/SYNASC.2016.020
Huang, Z. ; England, Matthew ; Davenport, J. H. ; Paulson, L. C. / Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases. 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. California : IEEE Computer Society, 2017. pp. 45-52
@inproceedings{759a929a53b94f5bb82a91982c442f7f,
title = "Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases",
abstract = "Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.",
keywords = "preconditioning, machine learning, support vector machine, computer algebra, cylindrical algebraic decomposition, groebner bases, Computers, Algebra, Machine learning algorithms, Support vector machines, Measurement, Electronic mail, Complexity theory",
author = "Z. Huang and Matthew England and Davenport, {J. H.} and Paulson, {L. C.}",
note = "{\circledC} 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. ISSN 2470-881X",
year = "2017",
month = "1",
day = "26",
doi = "10.1109/SYNASC.2016.020",
language = "English",
isbn = "978-1-5090-5707-8",
pages = "45--52",
booktitle = "18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing",
publisher = "IEEE Computer Society",

}

TY - GEN

T1 - Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases

AU - Huang, Z.

AU - England, Matthew

AU - Davenport, J. H.

AU - Paulson, L. C.

N1 - © 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. ISSN 2470-881X

PY - 2017/1/26

Y1 - 2017/1/26

N2 - Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.

AB - Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.

KW - preconditioning

KW - machine learning

KW - support vector machine

KW - computer algebra

KW - cylindrical algebraic decomposition

KW - groebner bases

KW - Computers

KW - Algebra

KW - Machine learning algorithms

KW - Support vector machines

KW - Measurement

KW - Electronic mail

KW - Complexity theory

U2 - 10.1109/SYNASC.2016.020

DO - 10.1109/SYNASC.2016.020

M3 - Conference proceeding

SN - 978-1-5090-5707-8

SP - 45

EP - 52

BT - 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

PB - IEEE Computer Society

CY - California

ER -