Real root finding for low rank linear matrices
We consider m × s matrices (with m ≥ s ) in a real affine subspace of dimension n . The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, ba...
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| Veröffentlicht in: | Applicable algebra in engineering, communication and computing Jg. 31; H. 2; S. 101 - 133 |
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| Sprache: | Englisch |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.03.2020
Springer Nature B.V Springer Verlag |
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| Abstract | We consider
m
×
s
matrices (with
m
≥
s
) in a real affine subspace of dimension
n
. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in
n
+
m
(
s
-
r
)
n
. It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach. |
|---|---|
| AbstractList | We consider m×s matrices (with m≥s) in a real affine subspace of dimension n. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in n+m(s-r)n. It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach. The problem of finding m × s matrices (with m ≥ s) of rank r in a real affine subspace of dimension n has many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is essentially polynomial in n+m(s−r) ; it improves on the state-of-the-art in the field. Moreover, computer experiments show the practical efficiency of our approach. We consider m × s matrices (with m ≥ s ) in a real affine subspace of dimension n . The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in n + m ( s - r ) n . It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach. |
| Author | Naldi, Simone Henrion, Didier Din, Mohab Safey El |
| Author_xml | – sequence: 1 givenname: Didier surname: Henrion fullname: Henrion, Didier organization: CNRS, LAAS, Université de Toulouse; LAAS, Faculty of Electrical Engineering, Czech Technical University in Prague – sequence: 2 givenname: Simone surname: Naldi fullname: Naldi, Simone email: simone.naldi@unilim.fr organization: CNRS, XLIM, UMR 7252, University of Limoges – sequence: 3 givenname: Mohab Safey El surname: Din fullname: Din, Mohab Safey El organization: CNRS, Inria, Laboratoire d’Informatique de Paris 6, Sorbonne Université |
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| Cites_doi | 10.1006/jcom.2000.0571 10.1007/3-540-33099-2 10.1007/s10208-008-9024-2 10.1007/s10208-014-9214-z 10.1007/978-3-642-15582-6_17 10.1145/2755996.2756667 10.1016/j.jsc.2013.03.004 10.1006/jsco.1993.1051 10.1145/2996450 10.1145/345542.345633 10.1080/10556788.2017.1341505 10.1007/978-3-540-69392-5 10.1525/9780520348097 10.1007/s10208-017-9361-0 10.1016/S0747-7171(88)80005-1 10.1016/j.jsc.2012.05.008 10.1016/j.jsc.2017.08.001 10.1007/s002000050114 10.1137/130931308 10.1007/978-1-4757-2189-8 10.1145/860854.860901 10.1007/3-540-51082-6_83 10.1137/15M1036543 10.1137/13094520X 10.1007/978-1-4612-5350-1 10.1145/2930889.2930916 10.1007/PL00004896 10.1016/j.jsc.2015.06.010 10.3792/chmm/1263317740 |
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| Keywords | 13-XX 68W30 Polynomial system solving 14Q20 12Y05 Low rank matrices Real algebraic geometry Symbolic computation |
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| References | FaugèreJean-CharlesFGb: A Library for Computing Gröbner BasesMathematical Software – ICMS 20102010Berlin, HeidelbergSpringer Berlin Heidelberg848710.1007/978-3-642-15582-6_17 GreuetASafey El DinMProbabilistic algorithm for the global optimization of a polynomial over a real algebraic setSIAM J. Optim.201424313131343324804310.1137/130931308 HarrisJAlgebraic Geometry: A First Course1992BerlinSpringer10.1007/978-1-4757-2189-8 Henrion, D., Naldi, S., Safey El Din, M.: Real root finding for rank defects in linear Hankel matrices. In: Proceedings of the 40th International Symposium on Symbolic and Algebraic Computation, Bath (UK), pp. 221–228 (2015) GiustiMLecerfGSalvyBA Gröbner-free alternative for polynomial system solvingJ. Complex.200117115421110.1006/jcom.2000.0571 TarskiAA Decision Method for Elementary Algebra and Geometry1951CaliforniaUniversity of California Press0044.25102 PoteauxASchostÉOn the complexity of computing with zero-dimensional triangular setsJ. Symb. Comput.201350110138299687110.1016/j.jsc.2012.05.008 Bonnard, B., Faugère, J-C., Jacquemard, A., Safey El Din, M., Verron, T.: Determinantal sets, singularities and application to optimal control in medical imagery. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC ’16, pp. 103–110, ACM, New York (2016) GrigorievDVorobjovNSolving systems of polynomial inequalities in subexponential timeJ. Symb. Comput.198851/2376494911210.1016/S0747-7171(88)80005-1 RouillierFSolving zero-dimensional systems through the rational univariate representationAppl. Algebra Eng. Commun. Comput.199995433461169717910.1007/s002000050114 EisenbudDCommutative Algebra with a View Toward Algebraic Geometry1995BerlinSpringer0819.13001 HenrionDNaldiSSafey El DinMSpectra: a Maple library for solving linear matrix inequalities in exact arithmeticOptim. Methods Softw.20193416278389466810.1080/10556788.2017.1341505 BankBGiustiMHeintzJLecerfGMateraGSolernóPDegeneracy loci and polynomial equation solvingFound. Comput. Math.2015151159184330369410.1007/s10208-014-9214-z HenrionDNaldiSSafey El DinMReal root finding for determinants of linear matricesJ. Symb. Comput.201574205238342404010.1016/j.jsc.2015.06.010 JeronimoGMateraGSolernóPWaissbeinADeformation techniques for sparse systemsFound. Comput. Math.200991150247228610.1007/s10208-008-9024-2 KroneckerLGrundzüge einer arithmetischen theorie der algebraischen GrössenJ. für die Reine und angewandte Math.1882921122157989614.0038.02 FaugèreJ-CSafey El DinMSpaenlehauerP-JOn the complexity of the generalized minrank problemJ. Symb. Comput.2013553058304265910.1016/j.jsc.2013.03.004 FaugèreJ-CGianniPLazardDMoraTEfficient computation of zero-dimensional Gröbner bases by change of orderingJ. Symb. Comput.199316432934410.1006/jsco.1993.1051 GianniPatriziaMoraTeoAlgebrric solution of systems of polynomirl equations using Groebher basesApplied Algebra, Algebraic Algorithms and Error-Correcting Codes1989Berlin, HeidelbergSpringer Berlin Heidelberg24725710.1007/3-540-51082-6_83 AbsilP-AMahonyRSepulchreROptimization Algorithms on Matrix Manifolds2009PrincetonPrinceton University Press1147.65043 Safey El Din, M.: Raglib (Real Algebraic Geometry library), Maple package (2007) Safey El DinMSchostÉBit complexity for multi-homogeneous polynomial system solving: application to polynomial minimizationJ. Symb. Comput.201887176206374434510.1016/j.jsc.2017.08.001 HenrionDNaldiSSafey El DinMExact algorithms for linear matrix inequalitiesSIAM J. Optim.201626425122539357459010.1137/15M1036543 Safey El Din, M., Schost, E.: A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets. J. ACM 63(48), (2017) OttavianiGSpaenlehauerP-JSturmfelsBExact solutions in structured low-rank approximationSIAM J. Matrix Anal. Appl.201435415211542328667910.1137/13094520X RanestadKristianAlgebraic Degree in Semidefinite and Polynomial OptimizationHandbook on Semidefinite, Conic and Polynomial Optimization2011Boston, MASpringer US61751334.90114 LasserreJ-BMoments, Positive Polynomials and Their Applications2010LondonImperial College Press1211.90007 Safey El Din, M., Schost, É.: Polar varieties and computation of one point in each connected component of a smooth real algebraic set. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ISSAC ’03, pp. 224–231, New York, NY. ACM (2003) ShafarevichIBasic Algebraic Geometry 11977BerlinSpringer BankBGiustiMHeintzJMbakopG-MPolar varieties and efficient real eliminationMath. Z.20012381115144186073810.1007/PL00004896 MacaulayFSThe Algebraic Theory of Modular Systems1916CambridgeCambridge University Press46.0167.01 BasuSPollackRRoyM-FAlgorithms in Real Algebraic in Mathematics20062BerlinSpringer10.1007/3-540-33099-2 ArbarelloECornalbaMGriffithsPAGeometry of Algebraic Curves: Volume II with a Contribution by Joseph Daniel Harris2011BerlinSpringer10.1007/978-3-540-69392-5 Lecerf, G.: Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions. In: Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, ISSAC ’00, pp. 209–216, New York. ACM (2000) CollinsGeorge E.Quantifier elimination for real closed fields by cylindrical algebraic decompostionLecture Notes in Computer Science1975Berlin, HeidelbergSpringer Berlin Heidelberg134183 KileelJKukelovaZPajdlaTSturmfelsBDistortion varietiesFound. Comput. Math.201818410431071383364910.1007/s10208-017-9361-0 HartshorneRAlgebraic Geometry2013BerlinSpringer0367.14001 396_CR34 B Bank (396_CR3) 2015; 15 396_CR33 396_CR32 B Bank (396_CR4) 2001; 238 J-C Faugère (396_CR11) 2013; 55 D Henrion (396_CR18) 2015; 74 George E. Collins (396_CR7) 1975 Patrizia Gianni (396_CR12) 1989 G Jeronimo (396_CR22) 2009; 9 D Grigoriev (396_CR15) 1988; 5 Jean-Charles Faugère (396_CR9) 2010 D Eisenbud (396_CR8) 1995 E Arbarello (396_CR2) 2011 L Kronecker (396_CR24) 1882; 92 A Greuet (396_CR14) 2014; 24 R Hartshorne (396_CR17) 2013 I Shafarevich (396_CR36) 1977 FS Macaulay (396_CR27) 1916 396_CR26 G Ottaviani (396_CR28) 2014; 35 A Poteaux (396_CR29) 2013; 50 J-B Lasserre (396_CR25) 2010 A Tarski (396_CR37) 1951 D Henrion (396_CR20) 2016; 26 F Rouillier (396_CR31) 1999; 9 S Basu (396_CR5) 2006 J Kileel (396_CR23) 2018; 18 M Giusti (396_CR13) 2001; 17 M Safey El Din (396_CR35) 2018; 87 Kristian Ranestad (396_CR30) 2011 P-A Absil (396_CR1) 2009 D Henrion (396_CR21) 2019; 34 396_CR19 J-C Faugère (396_CR10) 1993; 16 396_CR6 J Harris (396_CR16) 1992 |
| References_xml | – reference: GianniPatriziaMoraTeoAlgebrric solution of systems of polynomirl equations using Groebher basesApplied Algebra, Algebraic Algorithms and Error-Correcting Codes1989Berlin, HeidelbergSpringer Berlin Heidelberg24725710.1007/3-540-51082-6_83 – reference: RanestadKristianAlgebraic Degree in Semidefinite and Polynomial OptimizationHandbook on Semidefinite, Conic and Polynomial Optimization2011Boston, MASpringer US61751334.90114 – reference: LasserreJ-BMoments, Positive Polynomials and Their Applications2010LondonImperial College Press1211.90007 – reference: BankBGiustiMHeintzJLecerfGMateraGSolernóPDegeneracy loci and polynomial equation solvingFound. Comput. Math.2015151159184330369410.1007/s10208-014-9214-z – reference: AbsilP-AMahonyRSepulchreROptimization Algorithms on Matrix Manifolds2009PrincetonPrinceton University Press1147.65043 – reference: HenrionDNaldiSSafey El DinMExact algorithms for linear matrix inequalitiesSIAM J. Optim.201626425122539357459010.1137/15M1036543 – reference: FaugèreJean-CharlesFGb: A Library for Computing Gröbner BasesMathematical Software – ICMS 20102010Berlin, HeidelbergSpringer Berlin Heidelberg848710.1007/978-3-642-15582-6_17 – reference: Bonnard, B., Faugère, J-C., Jacquemard, A., Safey El Din, M., Verron, T.: Determinantal sets, singularities and application to optimal control in medical imagery. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC ’16, pp. 103–110, ACM, New York (2016) – reference: Safey El Din, M., Schost, É.: Polar varieties and computation of one point in each connected component of a smooth real algebraic set. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ISSAC ’03, pp. 224–231, New York, NY. ACM (2003) – reference: HartshorneRAlgebraic Geometry2013BerlinSpringer0367.14001 – reference: EisenbudDCommutative Algebra with a View Toward Algebraic Geometry1995BerlinSpringer0819.13001 – reference: HarrisJAlgebraic Geometry: A First Course1992BerlinSpringer10.1007/978-1-4757-2189-8 – reference: JeronimoGMateraGSolernóPWaissbeinADeformation techniques for sparse systemsFound. Comput. Math.200991150247228610.1007/s10208-008-9024-2 – reference: BankBGiustiMHeintzJMbakopG-MPolar varieties and efficient real eliminationMath. Z.20012381115144186073810.1007/PL00004896 – reference: OttavianiGSpaenlehauerP-JSturmfelsBExact solutions in structured low-rank approximationSIAM J. Matrix Anal. Appl.201435415211542328667910.1137/13094520X – reference: RouillierFSolving zero-dimensional systems through the rational univariate representationAppl. Algebra Eng. Commun. Comput.199995433461169717910.1007/s002000050114 – reference: Safey El Din, M.: Raglib (Real Algebraic Geometry library), Maple package (2007) – reference: HenrionDNaldiSSafey El DinMSpectra: a Maple library for solving linear matrix inequalities in exact arithmeticOptim. Methods Softw.20193416278389466810.1080/10556788.2017.1341505 – reference: TarskiAA Decision Method for Elementary Algebra and Geometry1951CaliforniaUniversity of California Press0044.25102 – reference: GrigorievDVorobjovNSolving systems of polynomial inequalities in subexponential timeJ. Symb. Comput.198851/2376494911210.1016/S0747-7171(88)80005-1 – reference: HenrionDNaldiSSafey El DinMReal root finding for determinants of linear matricesJ. Symb. Comput.201574205238342404010.1016/j.jsc.2015.06.010 – reference: FaugèreJ-CGianniPLazardDMoraTEfficient computation of zero-dimensional Gröbner bases by change of orderingJ. Symb. Comput.199316432934410.1006/jsco.1993.1051 – reference: CollinsGeorge E.Quantifier elimination for real closed fields by cylindrical algebraic decompostionLecture Notes in Computer Science1975Berlin, HeidelbergSpringer Berlin Heidelberg134183 – reference: KileelJKukelovaZPajdlaTSturmfelsBDistortion varietiesFound. Comput. Math.201818410431071383364910.1007/s10208-017-9361-0 – reference: Safey El DinMSchostÉBit complexity for multi-homogeneous polynomial system solving: application to polynomial minimizationJ. Symb. Comput.201887176206374434510.1016/j.jsc.2017.08.001 – reference: BasuSPollackRRoyM-FAlgorithms in Real Algebraic in Mathematics20062BerlinSpringer10.1007/3-540-33099-2 – reference: ArbarelloECornalbaMGriffithsPAGeometry of Algebraic Curves: Volume II with a Contribution by Joseph Daniel Harris2011BerlinSpringer10.1007/978-3-540-69392-5 – reference: Lecerf, G.: Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions. In: Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, ISSAC ’00, pp. 209–216, New York. ACM (2000) – reference: FaugèreJ-CSafey El DinMSpaenlehauerP-JOn the complexity of the generalized minrank problemJ. Symb. Comput.2013553058304265910.1016/j.jsc.2013.03.004 – reference: GreuetASafey El DinMProbabilistic algorithm for the global optimization of a polynomial over a real algebraic setSIAM J. Optim.201424313131343324804310.1137/130931308 – reference: Safey El Din, M., Schost, E.: A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets. J. ACM 63(48), (2017) – reference: MacaulayFSThe Algebraic Theory of Modular Systems1916CambridgeCambridge University Press46.0167.01 – reference: PoteauxASchostÉOn the complexity of computing with zero-dimensional triangular setsJ. Symb. 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| Snippet | We consider
m
×
s
matrices (with
m
≥
s
) in a real affine subspace of dimension
n
. The problem of finding elements of low rank in such spaces finds many... We consider m×s matrices (with m≥s) in a real affine subspace of dimension n. The problem of finding elements of low rank in such spaces finds many... The problem of finding m × s matrices (with m ≥ s) of rank r in a real affine subspace of dimension n has many applications in information and systems theory,... |
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| SubjectTerms | Algebraic Geometry Algorithms Artificial Intelligence Computer algebra Computer Hardware Computer Science Mathematical analysis Mathematics Matrix methods Original Paper Parameterization Polynomials Symbolic and Algebraic Manipulation Symbolic Computation System theory Systems theory Theory of Computation |
| Title | Real root finding for low rank linear matrices |
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