Probabilistic Analysis of the Grassmann Condition Number

We analyze the probability that a random m -dimensional linear subspace of both intersects a regular closed convex cone and lies within distance α of an m -dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone...

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Published in	Foundations of computational mathematics Vol. 15; no. 1; pp. 3 - 51
Main Authors	Amelunxen, Dennis, Bürgisser, Peter
Format	Journal Article
Language	English
Published	Boston Springer US 01.02.2015 Springer Springer Nature B.V
Subjects	Analysis Applications of Mathematics Computational mathematics Computer Science Convex programming Distribution (Probability theory) Economics Feasibility Foundations Linear and Multilinear Algebras Linear programming Manifolds Manifolds (Mathematics) Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Number theory Numerical Analysis Origins Perturbation (Mathematics) Probabilistic analysis Probability Subspaces Texts Topological manifolds United States 60D05 Tube formulas Spherically convex sets 52A22 52A55 Perturbation Convex programming Average analysis 90C31 Grassmann manifold 90C25 90C22 Condition number
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ISSN	1615-3375 1615-3383
DOI	10.1007/s10208-013-9178-4

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Abstract	We analyze the probability that a random m -dimensional linear subspace of both intersects a regular closed convex cone and lies within distance α of an m -dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C . This allows us to perform an average analysis of the Grassmann condition number for the homogeneous convex feasibility problem ∃ x ∈ C ∖0: Ax =0. The Grassmann condition number is a geometric version of Renegar’s condition number, which we have introduced recently in Amelunxen and Bürgisser (SIAM J. Optim. 22(3):1029–1041 ( 2012 )). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of are chosen i.i.d. standard normal, then for any regular cone C , we have . The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.
AbstractList	We analyze the probability that a random m-dimensional linear subspace of [R.sup.n] both intersects a regular closed convex cone C [subset or equal to] [R.sup.n] and lies within distance α of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number C(A) for the homogeneous convex feasibility problem ∃x ∈ C \ 0 : Ax = 0. The Grassmann condition number is a geometric version of Renegar's condition number, which we have introduced recently in Amelunxen and Burgisser (SIAM J. Optim. 22(3):1029-1041 (2012)). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of A ∈ [R.sup.mxn] are chosen i.i.d. standard normal, then for any regular cone C, we have E[lnC(A)] < 1.5ln(n) + 1.5. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds. We analyze the probability that a random m -dimensional linear subspace of both intersects a regular closed convex cone and lies within distance α of an m -dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C . This allows us to perform an average analysis of the Grassmann condition number for the homogeneous convex feasibility problem ∃ x ∈ C ∖0: Ax =0. The Grassmann condition number is a geometric version of Renegar’s condition number, which we have introduced recently in Amelunxen and Bürgisser (SIAM J. Optim. 22(3):1029–1041 ( 2012 )). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of are chosen i.i.d. standard normal, then for any regular cone C , we have . The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We analyze the probability that a random m-dimensional linear subspace of ... both intersects a regular closed convex cone ... and lies within distance [alpha] of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number [InlineEquation not available: see fulltext.] for the homogeneous convex feasibility problem xC0:Ax=0. The Grassmann condition number is a geometric version of Renegar's condition number, which we have introduced recently in Amelunxen and Bürgisser (SIAM J. Optim. 22(3):1029-1041 ( 2012 )). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of ... are chosen i.i.d. standard normal, then for any regular cone C, we have [InlineEquation not available: see fulltext.]. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We analyze the probability that a random m-dimensional linear subspace of ... both intersects a regular closed convex cone ... and lies within distance alpha of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number [InlineEquation not available: see fulltext.] for the homogeneous convex feasibility problem xC0:Ax=0. The Grassmann condition number is a geometric version of Renegar's condition number, which we have introduced recently in Amelunxen and Buergisser (SIAM J. Optim. 22(3):1029-1041 (2012)). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of ... are chosen i.i.d. standard normal, then for any regular cone C, we have [InlineEquation not available: see fulltext.]. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.
Audience	Academic
Author	Bürgisser, Peter Amelunxen, Dennis
Author_xml	– sequence: 1 givenname: Dennis surname: Amelunxen fullname: Amelunxen, Dennis email: damelunx@gmail.com organization: School of Mathematics, The University of Manchester – sequence: 2 givenname: Peter surname: Bürgisser fullname: Bürgisser, Peter organization: Institut für Mathematik, Technische Universität Berlin
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Cites_doi	10.1007/BF01581690 10.1016/j.jco.2007.01.002 10.1007/978-3-642-93014-0 10.1007/978-1-4757-2201-7 10.1017/CBO9780511804441 10.1007/s101070100237 10.56021/9781421407944 10.1137/S1052623494268467 10.1007/s101070050001 10.1007/978-3-642-38896-5 10.1017/CBO9780511526282 10.1137/S105262349223352X 10.1287/moor.1040.0120 10.1137/110835177 10.1017/CBO9780511616822 10.1007/s101070050088 10.1137/S1052623497323674 10.1007/BF01582132 10.1137/0805026 10.1137/S1052623401386794 10.1137/S105262349732829X 10.1007/978-3-540-78859-1 10.2307/2371513 10.1137/040616413 10.1090/S0025-5718-08-02060-7 10.1016/S0024-3795(03)00392-6 10.1007/s10107-007-0203-8 10.1090/S0025-5718-1988-0929546-7 10.1137/S1052623400373829
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Keywords	60D05 Tube formulas Spherically convex sets 52A22 52A55 Perturbation Convex programming Average analysis 90C31 Grassmann manifold 90C25 90C22 Condition number
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References	SchneiderR.WeilW.Stochastic and Integral Geometry2008BerlinSpringer10.1007/978-3-540-78859-11175.60003 BoothbyW.M.An Introduction to Differentiable Manifolds and Riemannian Geometry19862OrlandoAcademic Press0596.53001 VeraJ.R.On the complexity of linear programming under finite precision arithmeticMath. Program., Ser. A19988019112310.1007/BF015821320894.901121487309 BonnesenT.FenchelW.Theorie der Konvexen Körper1974BerlinSpringer10.1007/978-3-642-93014-00277.52001Berichtigter Reprint HelgasonS.Differential Geometry, Lie Groups, and Symmetric Spaces1978New YorkAcademic Press/Harcourt Brace Jovanovich0451.53038 SmaleS.Complexity theory and numerical analysisActa Numerica1997CambridgeCambridge University Press523551 AmelunxenD.BürgisserP.A coordinate-free condition number for convex programmingSIAM J. Optim.20122231029104110.1137/1108351771266.901393023762 ChenZ.DongarraJ.J.Condition numbers of Gaussian random matricesSIAM J. Matrix Anal. Appl.200527360362010.1137/0406164132208323(electronic) ThorpeJ.A.Elementary Topics in Differential Geometry1994New YorkSpringer PeñaJ.RenegarJ.Computing approximate solutions for convex conic systems of constraintsMath. Program., Ser. A200087335138310.1007/s1010700500011002.90039 SpivakM.Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus1965New York/AmsterdamBenjamin0141.05403 DemmelJ.W.The probability that a numerical analysis problem is difficultMath. Comp.19885044948010.1090/S0025-5718-1988-0929546-70657.65066929546 S. Glasauer, Integralgeometrie konvexer Körper im sphärischen Raum, Thesis, Univ. Freiburg i. Br. (1995). HornR.A.JohnsonC.R.Matrix Analysis1990CambridgeCambridge University Press0704.15002Corrected reprint of the 1985 original BürgisserP.CuckerF.LotzM.The probability that a slightly perturbed numerical analysis problem is difficultMath. Comp.2008772631559158310.1090/S0025-5718-08-02060-71195.650182398780 ChavelI.Riemannian Geometry20062CambridgeCambridge University Press10.1017/CBO97805116168221099.53001A modern introduction FreundR.M.VeraJ.R.Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithmSIAM J. Optim.199910115517610.1137/S105262349732829X0953.900441722119(electronic) BoydS.VandenbergheL.Convex Optimization2004CambridgeCambridge University Press10.1017/CBO97805118044411058.90049 RenegarJ.Linear programming, complexity theory and elementary functional analysisMath. Program., Ser. A19957032793510855.900851361602 WeylH.On the volume of tubesAm. J. Math.193961246147210.2307/23715131507388 D. Amelunxen, P. Bürgisser, Intrinsic volumes of symmetric cones, arXiv:1205.1863. BürgisserP.CuckerF.Condition: The Geometry of Numerical Algorithms2013BerlinSpringer SchneiderR.Convex Bodies: The Brunn-Minkowski Theory1993CambridgeCambridge University Press10.1017/CBO97805115262820798.52001 FreundR.M.VeraJ.R.Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear systemMath. Program., Ser. A199986222526010.1007/s1010700500880966.900481725230 BelloniA.FreundR.M.A geometric analysis of Renegar’s condition number, and its interplay with conic curvatureMath. Program., Ser. A200911919510710.1007/s10107-007-0203-81163.900292485489 SpivakM.A Comprehensive Introduction to Differential Geometry. Vol. I19993HoustonPublish or Perish EpelmanM.FreundR.M.A new condition measure, preconditioners, and relations between different measures of conditioning for conic linear systemsSIAM J. Optim.200212362765510.1137/S10526234003738291046.900381884909(electronic) RenegarJ.Incorporating condition measures into the complexity theory of linear programmingSIAM J. Optim.19955350652410.1137/08050260838.901391344668 GaoF.HugD.SchneiderR.Intrinsic volumes and polar sets in spherical spaceMath. Notae2003411591762049442(2001/2002). Homage to Luis Santaló. Vol. 1 (Spanish) BürgisserP.Smoothed analysis of condition numbersFoundations of Computational Mathematics2009CambridgeCambridge University Press141 FedererH.Geometric Measure Theory1969New YorkSpringer0176.00801 KobayashiS.NomizuK.Foundations of Differential Geometry1963New York/LondonInterscience/Wiley0119.37502 PeñaJ.Understanding the geometry of infeasible perturbations of a conic linear systemSIAM J. Optim.200010253455010.1137/S10526234973236740957.150051740958(electronic) VeraJ.C.RiveraJ.C.PeñaJ.HuiY.A primal-dual symmetric relaxation for homogeneous conic systemsJ. Complex.200723224526110.1016/j.jco.2007.01.0021118.65065 GolubG.H.Van LoanC.F.Matrix Computations20134BaltimoreJohns Hopkins University Press1268.65037 RenegarJ.Some perturbation theory for linear programmingMath. Program., Ser. A1994651739110.1007/BF015816900818.900731288040 FreundR.M.OrdóñezF.On an extension of condition number theory to nonconic convex optimizationMath. Oper. Res.200530117319410.1287/moor.1040.01201082.901512125143 GlasauerS.Integral geometry of spherically convex bodiesDiss. Summ. Math.199611–22192261420746 FilipowskiS.On the complexity of solving feasible linear programs specified with approximate dataSIAM J. Optim.1999941010104010.1137/S10526234942684670955.900861724774Dedicated to John E. Dennis Jr. on his 60th birthday MoszyńskaM.Selected Topics in Convex Geometry2006BostonBirkhäuser BostonTranslated and revised from the 2001 Polish original HowardR.The kinematic formula in Riemannian homogeneous spacesMem. Amer. Math. Soc.1993106509vi + 69 D. Amelunxen, Geometric analysis of the condition of the convex feasibility problem, Ph.D. thesis, Univ. Paderborn (2011). KlainD.A.RotaG.-C.Introduction to Geometric Probability1997CambridgeCambridge University Press0896.60004[Lincei Lectures] VeraJ.R.Ill-posedness and the complexity of deciding existence of solutions to linear programsSIAM J. Optim.19966354956910.1137/S105262349223352X0856.900771402192 StanleyR.P.Log-concave and unimodal sequences in algebra, combinatorics, and geometryGraph Theory and Its Applications: East and West1989New YorkN.Y. Acad. Sci.500535 do CarmoM.P.Riemannian Geometry1992BostonBirkhäuser Boston10.1007/978-1-4757-2201-70752.53001Translated from the second Portuguese edition by Francis Flaherty CheungD.CuckerF.A new condition number for linear programmingMath. Program., Ser. A20019111631741072.905641865268 MorganF.Geometric Measure Theory19952San DiegoAcademic Press0819.49024A beginner’s guide CuckerF.PeñaJ.A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machineSIAM J. Optim.2001/200212252255410.1137/S1052623401386794(electronic) PeñaJ.A characterization of the distance to infeasibility under block-structured perturbationsLinear Algebra Appl.200337019321610.1016/S0024-3795(03)00392-61028.150031994328 M. Spivak (9178_CR43) 1965 I. Chavel (9178_CR11) 2006 J.R. Vera (9178_CR48) 1998; 80 D. Cheung (9178_CR13) 2001; 91 J.C. Vera (9178_CR49) 2007; 23 P. Bürgisser (9178_CR10) 2008; 77 R. Schneider (9178_CR41) 2008 Z. Chen (9178_CR12) 2005; 27 J. Peña (9178_CR34) 2000; 10 9178_CR1 9178_CR2 J.W. Demmel (9178_CR15) 1988; 50 D. Amelunxen (9178_CR3) 2012; 22 R.A. Horn (9178_CR28) 1990 S. Boyd (9178_CR7) 2004 F. Morgan (9178_CR32) 1995 F. Cucker (9178_CR14) 2001/2002; 12 S. Smale (9178_CR42) 1997 P. Bürgisser (9178_CR8) 2009 H. Weyl (9178_CR50) 1939; 61 J.A. Thorpe (9178_CR46) 1994 D.A. Klain (9178_CR30) 1997 M.P. do Carmo (9178_CR16) 1992 F. Gao (9178_CR23) 2003; 41 S. Kobayashi (9178_CR31) 1963 R. Schneider (9178_CR40) 1993 S. Filipowski (9178_CR19) 1999; 9 M. Epelman (9178_CR17) 2002; 12 J. Peña (9178_CR36) 2000; 87 9178_CR24 R. Howard (9178_CR29) 1993; 106 M. Spivak (9178_CR44) 1999 J.R. Vera (9178_CR47) 1996; 6 A. Belloni (9178_CR4) 2009; 119 S. Glasauer (9178_CR25) 1996; 1 M. Moszyńska (9178_CR33) 2006 R.P. Stanley (9178_CR45) 1989 R.M. Freund (9178_CR22) 1999; 86 W.M. Boothby (9178_CR6) 1986 G.H. Golub (9178_CR26) 2013 R.M. Freund (9178_CR20) 2005; 30 R.M. Freund (9178_CR21) 1999; 10 T. Bonnesen (9178_CR5) 1974 S. Helgason (9178_CR27) 1978 J. Peña (9178_CR35) 2003; 370 J. Renegar (9178_CR38) 1995; 5 H. Federer (9178_CR18) 1969 P. Bürgisser (9178_CR9) 2013 J. Renegar (9178_CR37) 1994; 65 J. Renegar (9178_CR39) 1995; 70
References_xml	– reference: KobayashiS.NomizuK.Foundations of Differential Geometry1963New York/LondonInterscience/Wiley0119.37502 – reference: D. Amelunxen, Geometric analysis of the condition of the convex feasibility problem, Ph.D. thesis, Univ. Paderborn (2011). – reference: HowardR.The kinematic formula in Riemannian homogeneous spacesMem. Amer. Math. Soc.1993106509vi + 69 – reference: FreundR.M.VeraJ.R.Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear systemMath. Program., Ser. A199986222526010.1007/s1010700500880966.900481725230 – reference: FilipowskiS.On the complexity of solving feasible linear programs specified with approximate dataSIAM J. Optim.1999941010104010.1137/S10526234942684670955.900861724774Dedicated to John E. Dennis Jr. on his 60th birthday – reference: HornR.A.JohnsonC.R.Matrix Analysis1990CambridgeCambridge University Press0704.15002Corrected reprint of the 1985 original – reference: MorganF.Geometric Measure Theory19952San DiegoAcademic Press0819.49024A beginner’s guide – reference: BürgisserP.CuckerF.Condition: The Geometry of Numerical Algorithms2013BerlinSpringer – reference: SpivakM.A Comprehensive Introduction to Differential Geometry. Vol. I19993HoustonPublish or Perish – reference: BürgisserP.Smoothed analysis of condition numbersFoundations of Computational Mathematics2009CambridgeCambridge University Press141 – reference: RenegarJ.Linear programming, complexity theory and elementary functional analysisMath. Program., Ser. A19957032793510855.900851361602 – reference: FedererH.Geometric Measure Theory1969New YorkSpringer0176.00801 – reference: KlainD.A.RotaG.-C.Introduction to Geometric Probability1997CambridgeCambridge University Press0896.60004[Lincei Lectures] – reference: BoothbyW.M.An Introduction to Differentiable Manifolds and Riemannian Geometry19862OrlandoAcademic Press0596.53001 – reference: PeñaJ.Understanding the geometry of infeasible perturbations of a conic linear systemSIAM J. Optim.200010253455010.1137/S10526234973236740957.150051740958(electronic) – reference: SmaleS.Complexity theory and numerical analysisActa Numerica1997CambridgeCambridge University Press523551 – reference: ThorpeJ.A.Elementary Topics in Differential Geometry1994New YorkSpringer – reference: FreundR.M.OrdóñezF.On an extension of condition number theory to nonconic convex optimizationMath. Oper. Res.200530117319410.1287/moor.1040.01201082.901512125143 – reference: do CarmoM.P.Riemannian Geometry1992BostonBirkhäuser Boston10.1007/978-1-4757-2201-70752.53001Translated from the second Portuguese edition by Francis Flaherty – reference: S. Glasauer, Integralgeometrie konvexer Körper im sphärischen Raum, Thesis, Univ. Freiburg i. Br. (1995). – reference: PeñaJ.RenegarJ.Computing approximate solutions for convex conic systems of constraintsMath. Program., Ser. A200087335138310.1007/s1010700500011002.90039 – reference: RenegarJ.Some perturbation theory for linear programmingMath. Program., Ser. A1994651739110.1007/BF015816900818.900731288040 – reference: DemmelJ.W.The probability that a numerical analysis problem is difficultMath. Comp.19885044948010.1090/S0025-5718-1988-0929546-70657.65066929546 – reference: AmelunxenD.BürgisserP.A coordinate-free condition number for convex programmingSIAM J. Optim.20122231029104110.1137/1108351771266.901393023762 – reference: WeylH.On the volume of tubesAm. J. Math.193961246147210.2307/23715131507388 – reference: VeraJ.R.On the complexity of linear programming under finite precision arithmeticMath. Program., Ser. A19988019112310.1007/BF015821320894.901121487309 – reference: StanleyR.P.Log-concave and unimodal sequences in algebra, combinatorics, and geometryGraph Theory and Its Applications: East and West1989New YorkN.Y. Acad. Sci.500535 – reference: SchneiderR.Convex Bodies: The Brunn-Minkowski Theory1993CambridgeCambridge University Press10.1017/CBO97805115262820798.52001 – reference: VeraJ.C.RiveraJ.C.PeñaJ.HuiY.A primal-dual symmetric relaxation for homogeneous conic systemsJ. Complex.200723224526110.1016/j.jco.2007.01.0021118.65065 – reference: ChavelI.Riemannian Geometry20062CambridgeCambridge University Press10.1017/CBO97805116168221099.53001A modern introduction – reference: RenegarJ.Incorporating condition measures into the complexity theory of linear programmingSIAM J. Optim.19955350652410.1137/08050260838.901391344668 – reference: MoszyńskaM.Selected Topics in Convex Geometry2006BostonBirkhäuser BostonTranslated and revised from the 2001 Polish original – reference: HelgasonS.Differential Geometry, Lie Groups, and Symmetric Spaces1978New YorkAcademic Press/Harcourt Brace Jovanovich0451.53038 – reference: BoydS.VandenbergheL.Convex Optimization2004CambridgeCambridge University Press10.1017/CBO97805118044411058.90049 – reference: SpivakM.Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus1965New York/AmsterdamBenjamin0141.05403 – reference: BelloniA.FreundR.M.A geometric analysis of Renegar’s condition number, and its interplay with conic curvatureMath. Program., Ser. A200911919510710.1007/s10107-007-0203-81163.900292485489 – reference: PeñaJ.A characterization of the distance to infeasibility under block-structured perturbationsLinear Algebra Appl.200337019321610.1016/S0024-3795(03)00392-61028.150031994328 – reference: FreundR.M.VeraJ.R.Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithmSIAM J. Optim.199910115517610.1137/S105262349732829X0953.900441722119(electronic) – reference: SchneiderR.WeilW.Stochastic and Integral Geometry2008BerlinSpringer10.1007/978-3-540-78859-11175.60003 – reference: ChenZ.DongarraJ.J.Condition numbers of Gaussian random matricesSIAM J. Matrix Anal. Appl.200527360362010.1137/0406164132208323(electronic) – reference: GaoF.HugD.SchneiderR.Intrinsic volumes and polar sets in spherical spaceMath. Notae2003411591762049442(2001/2002). Homage to Luis Santaló. Vol. 1 (Spanish) – reference: D. Amelunxen, P. Bürgisser, Intrinsic volumes of symmetric cones, arXiv:1205.1863. – reference: GolubG.H.Van LoanC.F.Matrix Computations20134BaltimoreJohns Hopkins University Press1268.65037 – reference: VeraJ.R.Ill-posedness and the complexity of deciding existence of solutions to linear programsSIAM J. Optim.19966354956910.1137/S105262349223352X0856.900771402192 – reference: GlasauerS.Integral geometry of spherically convex bodiesDiss. Summ. Math.199611–22192261420746 – reference: BürgisserP.CuckerF.LotzM.The probability that a slightly perturbed numerical analysis problem is difficultMath. Comp.2008772631559158310.1090/S0025-5718-08-02060-71195.650182398780 – reference: CheungD.CuckerF.A new condition number for linear programmingMath. Program., Ser. A20019111631741072.905641865268 – reference: CuckerF.PeñaJ.A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machineSIAM J. Optim.2001/200212252255410.1137/S1052623401386794(electronic) – reference: EpelmanM.FreundR.M.A new condition measure, preconditioners, and relations between different measures of conditioning for conic linear systemsSIAM J. Optim.200212362765510.1137/S10526234003738291046.900381884909(electronic) – reference: BonnesenT.FenchelW.Theorie der Konvexen Körper1974BerlinSpringer10.1007/978-3-642-93014-00277.52001Berichtigter Reprint – volume: 65 start-page: 73 issue: 1 year: 1994 ident: 9178_CR37 publication-title: Math. Program., Ser. A doi: 10.1007/BF01581690 – volume-title: Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus year: 1965 ident: 9178_CR43 – volume: 23 start-page: 245 issue: 2 year: 2007 ident: 9178_CR49 publication-title: J. Complex. doi: 10.1016/j.jco.2007.01.002 – volume-title: Matrix Analysis year: 1990 ident: 9178_CR28 – volume-title: Theorie der Konvexen Körper year: 1974 ident: 9178_CR5 doi: 10.1007/978-3-642-93014-0 – volume-title: Riemannian Geometry year: 1992 ident: 9178_CR16 doi: 10.1007/978-1-4757-2201-7 – volume-title: Convex Optimization year: 2004 ident: 9178_CR7 doi: 10.1017/CBO9780511804441 – volume: 91 start-page: 163 issue: 1 year: 2001 ident: 9178_CR13 publication-title: Math. Program., Ser. A doi: 10.1007/s101070100237 – volume-title: Matrix Computations year: 2013 ident: 9178_CR26 doi: 10.56021/9781421407944 – volume-title: Geometric Measure Theory year: 1995 ident: 9178_CR32 – volume: 9 start-page: 1010 issue: 4 year: 1999 ident: 9178_CR19 publication-title: SIAM J. Optim. doi: 10.1137/S1052623494268467 – volume: 41 start-page: 159 year: 2003 ident: 9178_CR23 publication-title: Math. Notae – volume: 87 start-page: 351 issue: 3 year: 2000 ident: 9178_CR36 publication-title: Math. Program., Ser. A doi: 10.1007/s101070050001 – volume-title: Condition: The Geometry of Numerical Algorithms year: 2013 ident: 9178_CR9 doi: 10.1007/978-3-642-38896-5 – ident: 9178_CR2 – volume-title: Convex Bodies: The Brunn-Minkowski Theory year: 1993 ident: 9178_CR40 doi: 10.1017/CBO9780511526282 – volume-title: An Introduction to Differentiable Manifolds and Riemannian Geometry year: 1986 ident: 9178_CR6 – ident: 9178_CR24 – volume-title: Introduction to Geometric Probability year: 1997 ident: 9178_CR30 – volume-title: Differential Geometry, Lie Groups, and Symmetric Spaces year: 1978 ident: 9178_CR27 – volume-title: Selected Topics in Convex Geometry year: 2006 ident: 9178_CR33 – volume: 6 start-page: 549 issue: 3 year: 1996 ident: 9178_CR47 publication-title: SIAM J. Optim. doi: 10.1137/S105262349223352X – volume: 30 start-page: 173 issue: 1 year: 2005 ident: 9178_CR20 publication-title: Math. Oper. Res. doi: 10.1287/moor.1040.0120 – start-page: 1 volume-title: Foundations of Computational Mathematics year: 2009 ident: 9178_CR8 – volume-title: Elementary Topics in Differential Geometry year: 1994 ident: 9178_CR46 – ident: 9178_CR1 – volume: 70 start-page: 279 issue: 3 year: 1995 ident: 9178_CR39 publication-title: Math. Program., Ser. A – volume: 22 start-page: 1029 issue: 3 year: 2012 ident: 9178_CR3 publication-title: SIAM J. Optim. doi: 10.1137/110835177 – volume-title: Riemannian Geometry year: 2006 ident: 9178_CR11 doi: 10.1017/CBO9780511616822 – volume: 86 start-page: 225 issue: 2 year: 1999 ident: 9178_CR22 publication-title: Math. Program., Ser. A doi: 10.1007/s101070050088 – volume: 10 start-page: 534 issue: 2 year: 2000 ident: 9178_CR34 publication-title: SIAM J. Optim. doi: 10.1137/S1052623497323674 – volume: 1 start-page: 219 issue: 1–2 year: 1996 ident: 9178_CR25 publication-title: Diss. Summ. Math. – volume: 80 start-page: 91 issue: 1 year: 1998 ident: 9178_CR48 publication-title: Math. Program., Ser. A doi: 10.1007/BF01582132 – volume: 5 start-page: 506 issue: 3 year: 1995 ident: 9178_CR38 publication-title: SIAM J. Optim. doi: 10.1137/0805026 – volume: 12 start-page: 522 issue: 2 year: 2001/2002 ident: 9178_CR14 publication-title: SIAM J. Optim. doi: 10.1137/S1052623401386794 – volume: 10 start-page: 155 issue: 1 year: 1999 ident: 9178_CR21 publication-title: SIAM J. Optim. doi: 10.1137/S105262349732829X – volume-title: A Comprehensive Introduction to Differential Geometry. Vol. I year: 1999 ident: 9178_CR44 – volume: 106 start-page: vi + 69 issue: 509 year: 1993 ident: 9178_CR29 publication-title: Mem. Amer. Math. Soc. – start-page: 523 volume-title: Acta Numerica year: 1997 ident: 9178_CR42 – volume-title: Stochastic and Integral Geometry year: 2008 ident: 9178_CR41 doi: 10.1007/978-3-540-78859-1 – volume: 61 start-page: 461 issue: 2 year: 1939 ident: 9178_CR50 publication-title: Am. J. Math. doi: 10.2307/2371513 – volume: 27 start-page: 603 issue: 3 year: 2005 ident: 9178_CR12 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/040616413 – volume-title: Geometric Measure Theory year: 1969 ident: 9178_CR18 – start-page: 500 volume-title: Graph Theory and Its Applications: East and West year: 1989 ident: 9178_CR45 – volume: 77 start-page: 1559 issue: 263 year: 2008 ident: 9178_CR10 publication-title: Math. Comp. doi: 10.1090/S0025-5718-08-02060-7 – volume: 370 start-page: 193 year: 2003 ident: 9178_CR35 publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(03)00392-6 – volume: 119 start-page: 95 issue: 1 year: 2009 ident: 9178_CR4 publication-title: Math. Program., Ser. A doi: 10.1007/s10107-007-0203-8 – volume: 50 start-page: 449 year: 1988 ident: 9178_CR15 publication-title: Math. Comp. doi: 10.1090/S0025-5718-1988-0929546-7 – volume-title: Foundations of Differential Geometry year: 1963 ident: 9178_CR31 – volume: 12 start-page: 627 issue: 3 year: 2002 ident: 9178_CR17 publication-title: SIAM J. Optim. doi: 10.1137/S1052623400373829
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Snippet	We analyze the probability that a random m -dimensional linear subspace of both intersects a regular closed convex cone and lies within distance α of an m... We analyze the probability that a random m-dimensional linear subspace of [R.sup.n] both intersects a regular closed convex cone C [subset or equal to]... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We analyze the probability that a random m-dimensional linear subspace of ... both... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We analyze the probability that a random m-dimensional linear subspace of ... both...
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SubjectTerms	Analysis Applications of Mathematics Computational mathematics Computer Science Convex programming Distribution (Probability theory) Economics Feasibility Foundations Linear and Multilinear Algebras Linear programming Manifolds Manifolds (Mathematics) Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Number theory Numerical Analysis Origins Perturbation (Mathematics) Probabilistic analysis Probability Subspaces Texts Topological manifolds
Title	Probabilistic Analysis of the Grassmann Condition Number
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