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Volumn 57, Issue 12, 2009, Pages 4734-4743

An algorithm to compute averages on matrix lie groups

Author keywords

Averaging on curved spaces; Complex valued independent component analysis; Matrix Lie groups; Orthogonal group; Positive definite matrices; Unitary group

Indexed keywords

MATRIX LIE GROUPS; ORTHOGONAL GROUP; POSITIVE-DEFINITE MATRICES; RANDOM FLUCTUATION; SPECIAL ORTHOGONAL GROUP; STRUCTURED DATA; UNITARY GROUP;

EID: 70450270685     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2009.2027754     Document Type: Article
Times cited : (55)

References (40)
  • 1
    • 70450286289 scopus 로고    scopus 로고
    • Bi-Invariant Means in Lie Groups. Res. Rep. N. 5885 of the Institut National de Recherche en Informatique at en Automatique (INRIA), Apr.
    • V. Arsigny, X. Pennec, and N. Ayache, "Bi-Invariant Means in Lie Groups. Application to Left-Invariant Polyaffine Transformations," Res. Rep. N. 5885 of the Institut National de Recherche en Informatique at en Automatique (INRIA), Apr. 2006.
    • (2006) Application to Left-Invariant Polyaffine Transformations
    • Arsigny, V.1    Pennec, X.2    Ayache, N.3
  • 2
    • 0034136037 scopus 로고    scopus 로고
    • A fast fixed-point algorithm for independent component analysis of complex valued signals
    • E. Bingham and A. Hyvärinen, "A fast fixed-point algorithm for independent component analysis of complex valued signals," Int. J. Neural Syst., vol.10, no.1, pp. 1-8, 2000.
    • (2000) T. J. Neural Syst. , vol.10 , Issue.1 , pp. 1-8
    • Bingham, E.1    Hyvärinen, A.2
  • 3
    • 0003324926 scopus 로고
    • Topology and Geometry
    • 3rd ed. New York: Springer-Verlag, Jul.
    • G. E. Bredon, "Topology and Geometry," in Graduate Texts in Mathematics, 3rd ed. New York: Springer-Verlag, Jul. 1995, vol.139.
    • (1995) Graduate Texts in Mathematics , vol.139
    • Bredon, G.E.1
  • 4
    • 0003232417 scopus 로고    scopus 로고
    • Spherical averages and applications to spherical splines and interpolation
    • Apr.
    • S. R. Buss and J. P. Fillmore, "Spherical averages and applications to spherical splines and interpolation," ACM Trans. Graphics, vol.20, no.2, pp. 95-126, Apr. 2001.
    • (2001) ACM Trans. Graphics , vol.20 , Issue.2 , pp. 95-126
    • Buss, S.R.1    Fillmore, J.P.2
  • 5
    • 0030417779 scopus 로고    scopus 로고
    • Equivariant adaptive source separation
    • Dec.
    • J. F. Cardoso and B. Laheld, "Equivariant adaptive source separation," IEEE Trans. Signal Process., vol.44, no.12, pp. 3017-3030, Dec. 1996.
    • (1996) IEEE Trans. Signal Process. , vol.44 , Issue.12 , pp. 3017-3030
    • Cardoso, J.F.1    Laheld, B.2
  • 6
    • 1642544742 scopus 로고    scopus 로고
    • A class of intrinsic schemes for orthogonal integration
    • E. Celledoni and B. Owren, "A class of intrinsic schemes for orthogonal integration," SIAM J. Numer. Anal., vol.40, no.6, pp. 2069-2084, 2002.
    • (2002) SIAM J. Numer. Anal. , vol.40 , Issue.6 , pp. 2069-2084
    • Celledoni, E.1    Owren, B.2
  • 7
    • 0035537402 scopus 로고    scopus 로고
    • Approximating the logarithm of a matrix to specified accuracy
    • S.-H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, "Approximating the logarithm of a matrix to specified accuracy," SIAM J. Matrix Anal. Appl., vol.22, no.4, pp. 1112-1125, 2001.
    • (2001) SIAM J. Matrix Anal. Appl. , vol.22 , Issue.4 , pp. 1112-1125
    • Cheng, S.-H.1    Higham, N.J.2    Kenney, C.S.3    Laub, A.J.4
  • 9
    • 0000606459 scopus 로고
    • Complex ICA applied to the separation of radar signals
    • G. Desodt and D. Muller, "Complex ICA applied to the separation of radar signals," Proc. EUSIPCO, vol.I, pp. 665-668, 1990.
    • (1990) Proc. EUSIPCO , vol.1 , pp. 665-668
    • Desodt, G.1    Muller, D.2
  • 10
    • 0030519907 scopus 로고    scopus 로고
    • Computational techniques for real logarithms of matrices
    • L. Dieci, B. Morini, and A. Papini, "Computational techniques for real logarithms of matrices," SIAM J. Matrix Anal. Appl., vol.17, pp. 570-593, 1996.
    • (1996) SIAM J. Matrix Anal. Appl. , vol.17 , pp. 570-593
    • Dieci, L.1    Morini, B.2    Papini, A.3
  • 11
    • 0001586142 scopus 로고    scopus 로고
    • A theory for learning by weight flow on Stiefel-Grassman manifold
    • Jul.
    • S. Fiori, "A theory for learning by weight flow on Stiefel-Grassman manifold," Neural Comput., vol.13, no.7, pp. 1625-1647, Jul. 2001.
    • (2001) Neural Comput. , vol.13 , Issue.7 , pp. 1625-1647
    • Fiori, S.1
  • 12
    • 2442478521 scopus 로고    scopus 로고
    • A fast fixed-point neural blind deconvolution algorithm
    • Mar.
    • S. Fiori, "A fast fixed-point neural blind deconvolution algorithm," IEEE Trans. Neural Netw., vol.15, no.2, pp. 455-459, Mar. 2004.
    • (2004) IEEE Trans. Neural Netw. , vol.15 , Issue.2 , pp. 455-459
    • Fiori, S.1
  • 13
    • 21844443579 scopus 로고    scopus 로고
    • Quasi-geodesic neural learning algorithms over the orthogonal group: A tutorial
    • May
    • S. Fiori, "Quasi-geodesic neural learning algorithms over the orthogonal group: A tutorial," J. Mach. Learn. Res., vol.6, pp. 743-781, May 2005.
    • (2005) J. Mach. Learn. Res. , vol.6 , pp. 743-781
    • Fiori, S.1
  • 14
    • 17044396848 scopus 로고    scopus 로고
    • Non-linear complex-valued extensions of Hebbian learning: An essay
    • S. Fiori, "Non-linear complex-valued extensions of Hebbian learning: An essay," Neural Comput., vol.17, no.4, pp. 779-838, 2005.
    • (2005) Neural Comput. , vol.17 , Issue.4 , pp. 779-838
    • Fiori, S.1
  • 15
    • 34249877665 scopus 로고    scopus 로고
    • Learning independent components on the orthogonal group of matrices by retractions
    • Jun.
    • S. Fiori, "Learning independent components on the orthogonal group of matrices by retractions," Neural Process. Lett., vol.25, no.3, pp. 187-198, Jun. 2007.
    • (2007) Neural Process. Lett. , vol.25 , Issue.3 , pp. 187-198
    • Fiori, S.1
  • 16
    • 41849088853 scopus 로고    scopus 로고
    • A study on neural learning on manifold foliations: The case of the Lie group SU(3)
    • Apr.
    • S. Fiori, "A study on neural learning on manifold foliations: The case of the Lie group SU(3)," Neural Comput., vol.20, no.4, pp. 1091-1117, Apr. 2008.
    • (2008) Neural Comput. , vol.20 , Issue.4 , pp. 1091-1117
    • Fiori, S.1
  • 17
    • 44649088387 scopus 로고    scopus 로고
    • Leap-frog-type learning algorithms over the Lie group of unitary matrices
    • Jun.
    • S. Fiori, "Leap-frog-type learning algorithms over the Lie group of unitary matrices," Neurocomput.(Special Issue on Advances in Blind Signal Processing), vol.71, no.10-12, pp. 2224-2244, Jun. 2008.
    • (2008) Neurocomput.(Special Issue on Advances in Blind Signal Processing) , vol.71 , Issue.10-12 , pp. 2224-2244
    • Fiori, S.1
  • 18
    • 67349130556 scopus 로고    scopus 로고
    • On vector averaging over the unit hyphersphere
    • Jul.
    • S. Fiori, "On vector averaging over the unit hyphersphere," Digital Signal Process., vol.19, no.4, pp. 715-725, Jul. 2009.
    • (2009) Digital Signal Process. , vol.19 , Issue.4 , pp. 715-725
    • Fiori, S.1
  • 19
    • 33750164640 scopus 로고    scopus 로고
    • Riemannian geometry for the statistical analysis of diffusion tensor data
    • Feb.
    • P. T. Fletcher and S. Joshi, "Riemannian geometry for the statistical analysis of diffusion tensor data," Signal Process., vol.87, no.2, pp. 250-262, Feb. 2007.
    • (2007) Signal Process. , vol.87 , Issue.2 , pp. 250-262
    • Fletcher, P.T.1    Joshi, S.2
  • 20
    • 0000315591 scopus 로고
    • Les élements aléatoires de nature quelconque dans un espace distancié
    • M. Fréchet, "Les élements aléatoires de nature quelconque dans un espace distancié," Ann. l'Institut Henri Poincaré, vol.10, pp. 215-310, 1948.
    • (1948) Ann. l'Institut Henri Poincaré , vol.10 , pp. 215-310
    • Fréchet, M.1
  • 23
  • 24
    • 27144549311 scopus 로고    scopus 로고
    • The scaling and squaring method for the matrix exponential revisited
    • N. J. Higham, "The scaling and squaring method for the matrix exponential revisited," SIAM J. Matrix Anal. Appl., vol.26, pp. 1179-1193, 2005.
    • (2005) SIAM J. Matrix Anal. Appl. , vol.26 , pp. 1179-1193
    • Higham, N.J.1
  • 25
    • 0035578991 scopus 로고    scopus 로고
    • Uniqueness of matrix square roots and an application
    • C. R. Johnson, K. Okubo, and R. Reams, "Uniqueness of matrix square roots and an application," Linear Algebra Its Appl., vol.323, pp. 51-60, 2001.
    • (2001) Linear Algebra Its Appl. , vol.323 , pp. 51-60
    • Johnson, C.R.1    Okubo, K.2    Reams, R.3
  • 26
    • 84980162923 scopus 로고
    • Riemannian center of mass and mollifier smoothing
    • H. Karcher, "Riemannian center of mass and mollifier smoothing," Commun. Pure Appl. Math., vol.30, pp. 509-541, 1977.
    • (1977) Commun. Pure Appl. Math. , vol.30 , pp. 509-541
    • Karcher, H.1
  • 28
    • 59749092732 scopus 로고    scopus 로고
    • Stability of an elastic rod buckling into a soft wall
    • Aug.
    • R. S. Manning and G. B. Bulman, "Stability of an elastic rod buckling into a soft wall," Proc. Roy. Soc. A, Math., Phys. Eng. Sci., vol.461, no.2060, pp. 2423-2450, Aug. 2005.
    • (2005) Proc. Roy. Soc. A, Math., Phys. Eng. Sci. , vol.461 , Issue.2060 , pp. 2423-2450
    • Manning, R.S.1    Bulman, G.B.2
  • 29
    • 33747873784 scopus 로고    scopus 로고
    • A centroid (Karcher mean) approach to the joint approximate diagonalisation problem: The real symmetric case
    • J. H. Manton, "A centroid (Karcher mean) approach to the joint approximate diagonalisation problem: The real symmetric case," Digit. Signal Process., vol.16, pp. 468-478, 2006.
    • (2006) Digit. Signal Process. , vol.16 , pp. 468-478
    • Manton, J.H.1
  • 30
    • 0037281553 scopus 로고    scopus 로고
    • Means and averaging in the group of rotations
    • M. Moakher, "Means and averaging in the group of rotations," SIAM J. Matrix Anal. Appl., vol.24, no.1, pp. 1-16, 2002.
    • (2002) SIAM J. Matrix Anal. Appl. , vol.24 , Issue.1 , pp. 1-16
    • Moakher, M.1
  • 31
    • 22944439423 scopus 로고    scopus 로고
    • A differential geometry approach to the geometric mean of symmetric positive definite matrices
    • M. Moakher, "A differential geometry approach to the geometric mean of symmetric positive definite matrices," SIAM J. Matrix Anal. Appl., vol.26, pp. 735-747, 2005.
    • (2005) SIAM J. Matrix Anal. Appl. , vol.26 , pp. 735-747
    • Moakher, M.1
  • 32
    • 0009317622 scopus 로고
    • Some remarks on dynamical systems and numerical analysis
    • L. Lara-Carrero and J. Lewowicz, Eds. Caracas, Venezuela: Equinoccio
    • M. Shub, "Some remarks on dynamical systems and numerical analysis," in Dynamical Systems and Partial Differential Equations: Proceedings of the VII ELAM, L. Lara-Carrero and J. Lewowicz, Eds. Caracas, Venezuela: Equinoccio, 1986, pp. 62-92.
    • (1986) Dynamical Systems and Partial Differential Equations: Proceedings of the VII ELAM , pp. 62-92
    • Shub, M.1
  • 35
    • 18844440455 scopus 로고    scopus 로고
    • Covariance, subspace, and instrinsic Cramér-Rao bounds
    • May
    • S. T. Smith, "Covariance, subspace, and instrinsic Cramér-Rao bounds," IEEE Trans. Signal Process., vol.53, no.5, pp. 1610-1630, May 2005.
    • (2005) IEEE Trans. Signal Process. , vol.53 , Issue.5 , pp. 1610-1630
    • Smith, S.T.1
  • 36
    • 33947655308 scopus 로고    scopus 로고
    • Simultaneous tracking of the best basis in reduced- rank Wiener filter
    • Toulouse, France, May
    • T. Tanaka and S. Fiori, "Simultaneous tracking of the best basis in reduced- rank Wiener filter," in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Process. (ICASSP), Toulouse, France, May 2006, vol.III, pp. 548-551.
    • (2006) Proc. IEEE Int. Conf. Acoustics, Speech, Signal Process. (ICASSP) , vol.3 , pp. 548-551
    • Tanaka, T.1    Fiori, S.2
  • 37
    • 1842477730 scopus 로고    scopus 로고
    • Uniqueness of complex and multidimensional independent component analysis
    • F. Theis, "Uniqueness of complex and multidimensional independent component analysis," Signal Process., vol.84, no.5, pp. 951-956, 2004.
    • (2004) Signal Process , vol.84 , Issue.5 , pp. 951-956
    • Theis, F.1
  • 38
    • 21844471282 scopus 로고    scopus 로고
    • Matrix exponentiated gradient updates for on-line learning and Bregman projection
    • K. Tsuda, G. Rätsch, and M. K. Warmuth, "Matrix exponentiated gradient updates for on-line learning and Bregman projection," J. Mach. Learn. Res., vol.6, pp. 995-1018, 2005.
    • (2005) J. Mach. Learn. Res. , vol.6 , pp. 995-1018
    • Tsuda, K.1    Rätsch, G.2    Warmuth, M.K.3
  • 39
    • 0037401345 scopus 로고    scopus 로고
    • Computation of symmetric positive definite Toeplitz matrices by the hybrid steepest descent method
    • May
    • K. Slavakis, I. Yamada, and K. Sakaniwa, "Computation of symmetric positive definite Toeplitz matrices by the hybrid steepest descent method," Signal Process., vol.83, no.5, pp. 1135-1140, May 2003.
    • (2003) Signal Process , vol.83 , Issue.5 , pp. 1135-1140
    • Slavakis, K.1    Yamada, I.2    Sakaniwa, K.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.