메뉴 건너뛰기




Volumn , Issue , 2009, Pages 781-788

Tensors versus matrices usefulness and unexpected properties

Author keywords

Blind identification; Canonical decomposition; Deflation; Factor analysis; High order statistics; Low rank approximation; Parafac; Separation of variables; Tensor rank

Indexed keywords

BLIND IDENTIFICATIONS; CANONICAL DECOMPOSITION; FACTOR ANALYSIS; HIGH ORDER STATISTICS; LOW RANK APPROXIMATIONS; PARAFAC; SEPARATION OF VARIABLES;

EID: 72349095077     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/SSP.2009.5278471     Document Type: Conference Paper
Times cited : (19)

References (56)
  • 1
    • 0005332910 scopus 로고
    • Ranks of tensors and change of base field
    • G. M. BERGMAN, "Ranks of tensors and change of base field," Journal of Algebra, vol. 11, pp. 613-621, 1969.
    • (1969) Journal of Algebra , vol.11 , pp. 613-621
    • BERGMAN, G.M.1
  • 3
    • 0007725617 scopus 로고
    • Typical tensorial rank
    • T. LICKTEIG, "Typical tensorial rank," Linear Algebra Appl., vol. 69, pp. 95-120, 1985.
    • (1985) Linear Algebra Appl , vol.69 , pp. 95-120
    • LICKTEIG, T.1
  • 5
    • 35348895048 scopus 로고
    • Multiple invariants and generalized rank of a p-way matrix or tensor
    • F. L. HITCHCOCK, "Multiple invariants and generalized rank of a p-way matrix or tensor," J. Math. and Phys., vol. 7, no. 1, pp. 40-79, 1927.
    • (1927) J. Math. and Phys , vol.7 , Issue.1 , pp. 40-79
    • HITCHCOCK, F.L.1
  • 6
    • 0007772789 scopus 로고
    • Global properties of tensor rank
    • T. D. HOWELL, "Global properties of tensor rank," Linear Algebra and Applications, vol. 22, pp. 9-23, 1978.
    • (1978) Linear Algebra and Applications , vol.22 , pp. 9-23
    • HOWELL, T.D.1
  • 7
    • 48749142095 scopus 로고
    • Canonical forms for symmetric tensors
    • D. A. WEINSTEIN, "Canonical forms for symmetric tensors," Linear Algebra Appl., pp. 271-282, 1984.
    • (1984) Linear Algebra Appl , pp. 271-282
    • WEINSTEIN, D.A.1
  • 9
    • 72349092379 scopus 로고    scopus 로고
    • P. COMON, Tensor decompositions, state of the art and applications, in Mathematics in Signal Processing V, J. G. McWhirter and I. K. Proudler, Eds., pp. 1-24. Clarendon Press, Oxford, UK, 2002, arXiv:0905.0454v1.
    • P. COMON, "Tensor decompositions, state of the art and applications," in Mathematics in Signal Processing V, J. G. McWhirter and I. K. Proudler, Eds., pp. 1-24. Clarendon Press, Oxford, UK, 2002, arXiv:0905.0454v1.
  • 10
    • 33846606151 scopus 로고    scopus 로고
    • Singular values and eigenvalues of tensors: A variational approach
    • Puerto Vallarta, Mexico, 13-15 Dec
    • L-H. LIM, "Singular values and eigenvalues of tensors: a variational approach," in IEEE Int. Workshop on Comput. Adv. Multi-Sensor Adapt. Proc., Puerto Vallarta, Mexico, 13-15 Dec. 2005.
    • (2005) IEEE Int. Workshop on Comput. Adv. Multi-Sensor Adapt. Proc
    • LIM, L.-H.1
  • 11
    • 0032224093 scopus 로고    scopus 로고
    • Blind channel identification and extraction of more sources than sensors, keynote address
    • San Diego, July 19-24
    • P. COMON, "Blind channel identification and extraction of more sources than sensors," keynote address. in SPIE Conference, San Diego, July 19-24 1998, pp. 2-13,
    • (1998) SPIE Conference , pp. 2-13
    • COMON, P.1
  • 12
    • 33750796673 scopus 로고    scopus 로고
    • Rank and eigenvalues of a symmetric tensor, the multivariate homogeneous polynomial and the algebraic hyper-surface it defines
    • L. QI, "Rank and eigenvalues of a symmetric tensor, the multivariate homogeneous polynomial and the algebraic hyper-surface it defines," Jour.Symbolic Computation, vol. 41, pp. 1309-1327, 2006.
    • (2006) Jour.Symbolic Computation , vol.41 , pp. 1309-1327
    • QI, L.1
  • 13
    • 0001536630 scopus 로고
    • Apolarity and canonical forms for homogeneous polynomials
    • R. EHRENBORG and G. C. ROTA, "Apolarity and canonical forms for homogeneous polynomials," European Jour. Combinatorics, vol. 14, pp. 157-181, 1993.
    • (1993) European Jour. Combinatorics , vol.14 , pp. 157-181
    • EHRENBORG, R.1    ROTA, G.C.2
  • 14
    • 0001419778 scopus 로고
    • Inverse system of a symbolic power. II. the Waring problem for forms
    • A. IARROBINO, "Inverse system of a symbolic power. II. the Waring problem for forms," J. Algebra, vol. 174, pp. 1091-1110, 1995.
    • (1995) J. Algebra , vol.174 , pp. 1091-1110
    • IARROBINO, A.1
  • 15
    • 0034559666 scopus 로고    scopus 로고
    • The typical rank of tall three-way arrays
    • Sept
    • J. M. F. ten BERGE, "The typical rank of tall three-way arrays," Psychometrika, vol. 65, no. 5, pp. 525-532, Sept. 2000.
    • (2000) Psychometrika , vol.65 , Issue.5 , pp. 525-532
    • ten BERGE, J.M.F.1
  • 16
    • 64749092635 scopus 로고    scopus 로고
    • Generic and typical ranks of multi-way arrays
    • June
    • P. COMON, J. M. F. ten BERGE, L. De LATHAUWER, and J. CASTAING, "Generic and typical ranks of multi-way arrays," Linear Algebra Appl., vol. 430, no. 11-12, pp. 2997-3007, June 2009.
    • (2009) Linear Algebra Appl , vol.430 , Issue.11-12 , pp. 2997-3007
    • COMON, P.1    ten BERGE, J.M.F.2    De LATHAUWER, L.3    CASTAING, J.4
  • 17
    • 0030234706 scopus 로고    scopus 로고
    • Decomposition of quantics in sums of powers of linear forms
    • special issue on High-Order Statistics, Sept
    • P. COMON and B. MOURRAIN, "Decomposition of quantics in sums of powers of linear forms," Signal Processing, Elsevier, vol. 53, no. 2, pp. 93-107, Sept. 1996, special issue on High-Order Statistics.
    • (1996) Signal Processing, Elsevier , vol.53 , Issue.2 , pp. 93-107
    • COMON, P.1    MOURRAIN, B.2
  • 18
    • 70349656252 scopus 로고    scopus 로고
    • Tensor decompositions, alternating least squares and other tales
    • P. COMON, X. LUCIANI, and A. L. F. de ALMEIDA, "Tensor decompositions, alternating least squares and other tales," Jour. Chemometrics, 2009.
    • (2009) Jour. Chemometrics
    • COMON, P.1    LUCIANI, X.2    de ALMEIDA, A.L.F.3
  • 19
    • 0033653594 scopus 로고    scopus 로고
    • On the uniqueness of multilinear decomposition of N-way arrays
    • N. D. SIDIROPOULOS and R. BRO, "On the uniqueness of multilinear decomposition of N-way arrays," Jour. Chemo., vol. 14, pp. 229-239, 2000.
    • (2000) Jour. Chemo , vol.14 , pp. 229-239
    • SIDIROPOULOS, N.D.1    BRO, R.2
  • 20
    • 48749101457 scopus 로고
    • Three-way arrays: Rank and uniqueness of trilinear decompositions
    • J. B. KRUSKAL, "Three-way arrays: Rank and uniqueness of trilinear decompositions," Linear Algebra and Applications, vol. 18, pp. 95-138, 1977.
    • (1977) Linear Algebra and Applications , vol.18 , pp. 95-138
    • KRUSKAL, J.B.1
  • 21
    • 33746356349 scopus 로고    scopus 로고
    • A link between canonical decomposition in multilinear algebra and simultaneous matrix diagonalization
    • L. De LATHAUWER, "A link between canonical decomposition in multilinear algebra and simultaneous matrix diagonalization," SIAM Jour. Matrix Ana. Appl., vol. 28, no. 3, pp. 642-666, 2006.
    • (2006) SIAM Jour. Matrix Ana. Appl , vol.28 , Issue.3 , pp. 642-666
    • De LATHAUWER, L.1
  • 23
    • 0036042351 scopus 로고    scopus 로고
    • Computation of canonical forms for ternary cubics
    • ACM
    • I. A. KOGAN and M. M. MAZA, "Computation of canonical forms for ternary cubics," in ISSAC. 2000, pp. 151-160, ACM.
    • (2000) ISSAC , pp. 151-160
    • KOGAN, I.A.1    MAZA, M.M.2
  • 24
    • 72349090575 scopus 로고    scopus 로고
    • Subtracting a best rank-1 approximation does not necessarily decrease tensor rank
    • Glasgow, Scotland, Aug. 24-28
    • A. STEGEMAN and P. COMON, "Subtracting a best rank-1 approximation does not necessarily decrease tensor rank," in EUSIPCO'09, Glasgow, Scotland, Aug. 24-28 2009.
    • (2009) EUSIPCO'09
    • STEGEMAN, A.1    COMON, P.2
  • 25
    • 55349142218 scopus 로고    scopus 로고
    • Tensor rank and the ill-posedness of the best low-rank approximation problem
    • V. De SILVA and L-H. LIM, "Tensor rank and the ill-posedness of the best low-rank approximation problem," SIAM Jour. Matrix Ana. Appl., vol. 30, no. 3, pp. 1084-1127, 2008.
    • (2008) SIAM Jour. Matrix Ana. Appl , vol.30 , Issue.3 , pp. 1084-1127
    • De SILVA, V.1    LIM, L.-H.2
  • 26
    • 0000144406 scopus 로고
    • Algebraic Geometry, a first course
    • of, Springer
    • J. HARRIS, Algebraic Geometry, a first course, vol. 133 of Graduate Texts in Math., Springer, 1992.
    • (1992) Graduate Texts in Math , vol.133
    • HARRIS, J.1
  • 27
    • 2142824430 scopus 로고    scopus 로고
    • Analytical blind discrete source separation
    • Tampere, Finland, 5-8 sept
    • O. GRELLIER and P. COMON, "Analytical blind discrete source separation," in Eusipco, Tampere, Finland, 5-8 sept. 2000.
    • (2000) Eusipco
    • GRELLIER, O.1    COMON, P.2
  • 28
    • 0002883410 scopus 로고
    • Higher-order power method, application in Independent Component Analysis
    • Las Vegas, 10-14 Dec
    • L. de LATHAUWER, P. COMON, et al., "Higher-order power method, application in Independent Component Analysis," in NOLTA Conference, Las Vegas, 10-14 Dec 1995, vol. 1, pp. 91-96.
    • (1995) NOLTA Conference , vol.1 , pp. 91-96
    • de LATHAUWER, L.1    COMON, P.2
  • 29
    • 0036018642 scopus 로고    scopus 로고
    • One the best rank-1 approximation of higher-order supersymmetric tensors
    • E. KOFIDIS and P. REGALIA, "One the best rank-1 approximation of higher-order supersymmetric tensors," SIAM Jour. Matrix Ana. Appl., vol. 23, pp. 863-884, 2002.
    • (2002) SIAM Jour. Matrix Ana. Appl , vol.23 , pp. 863-884
    • KOFIDIS, E.1    REGALIA, P.2
  • 30
    • 0034144761 scopus 로고    scopus 로고
    • On the best rank-1 and rank-(R1,R2,⋯RN) approximation of high-order tensors
    • Apr
    • L. de LATHAUWER, B. de MOOR, and J. VANDEWALLE, "On the best rank-1 and rank-(R1,R2,⋯RN) approximation of high-order tensors," SIAM Jour. Matrix Ana. Appl., vol. 21, no. 4, pp. 1324-1342, Apr. 2000.
    • (2000) SIAM Jour. Matrix Ana. Appl , vol.21 , Issue.4 , pp. 1324-1342
    • de LATHAUWER, L.1    de MOOR, B.2    VANDEWALLE, J.3
  • 31
    • 70349696077 scopus 로고    scopus 로고
    • Nonnegative approximations of nonnegative tensors
    • to appear
    • L-H. LIM and P. COMON, "Nonnegative approximations of nonnegative tensors," Jour. Chemometrics, 2009, to appear.
    • (2009) Jour. Chemometrics
    • LIM, L.-H.1    COMON, P.2
  • 32
    • 0033653782 scopus 로고    scopus 로고
    • Construction and analysis of degenerate Parafac models
    • P. PAATERO, "Construction and analysis of degenerate Parafac models," Jour. Chemometrics, vol. 14, pp. 285-299, 2000.
    • (2000) Jour. Chemometrics , vol.14 , pp. 285-299
    • PAATERO, P.1
  • 33
    • 33645813863 scopus 로고
    • Approximate solutions for the bilinear form computational problem
    • D. BINI, G. LOTTI, and F. ROMANI, "Approximate solutions for the bilinear form computational problem," SIAM Jour. Comput., pp. 692-697, 1980.
    • (1980) SIAM Jour. Comput , pp. 692-697
    • BINI, D.1    LOTTI, G.2    ROMANI, F.3
  • 34
    • 0033275630 scopus 로고    scopus 로고
    • The multilinear engine: A table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model
    • Dec
    • P. PAATERO, "The multilinear engine: A table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model," Journal of Computational and Graphical Statistics, vol. 8, no. 4, pp. 854-888, Dec. 1999.
    • (1999) Journal of Computational and Graphical Statistics , vol.8 , Issue.4 , pp. 854-888
    • PAATERO, P.1
  • 35
    • 30144444694 scopus 로고    scopus 로고
    • A comparison of algorithms for fitting the Parafac model
    • G. TOMASI and R. BRO, "A comparison of algorithms for fitting the Parafac model," Comp. Stat. Data Anal., vol. 50, pp. 1700-1734, 2006.
    • (2006) Comp. Stat. Data Anal , vol.50 , pp. 1700-1734
    • TOMASI, G.1    BRO, R.2
  • 36
    • 0032563557 scopus 로고    scopus 로고
    • Improving the speed of multiway algorithms. Part II: Compression
    • R. BRO and C. A. ANDERSSON, "Improving the speed of multiway algorithms. Part II: Compression," Chemometrics and Intelligent Laboratory Systems, vol. 42, no. 1-2, pp. 105-113, 1998.
    • (1998) Chemometrics and Intelligent Laboratory Systems , vol.42 , Issue.1-2 , pp. 105-113
    • BRO, R.1    ANDERSSON, C.A.2
  • 39
    • 66849099206 scopus 로고    scopus 로고
    • Enhanced line search: A novel method to accelerate Parafac
    • Sept
    • M. RAJIH, P. COMON, and R. HARSHMAN, "Enhanced line search: A novel method to accelerate Parafac," SIAM Jour. Matrix Ana. Appl., vol. 30, no. 3, pp. 1148-1171, Sept. 2008.
    • (2008) SIAM Jour. Matrix Ana. Appl , vol.30 , Issue.3 , pp. 1148-1171
    • RAJIH, M.1    COMON, P.2    HARSHMAN, R.3
  • 40
    • 36049038610 scopus 로고    scopus 로고
    • An enhanced line search scheme for complex-valued tensor decompositions. application in ds-cdma
    • Mar
    • D. NION and L. DeLATHAUWER, "An enhanced line search scheme for complex-valued tensor decompositions. application in ds-cdma," Signal Processing, vol. 88, no. 3, pp. 749-755, Mar. 2008.
    • (2008) Signal Processing , vol.88 , Issue.3 , pp. 749-755
    • NION, D.1    DeLATHAUWER, L.2
  • 41
    • 68649114427 scopus 로고    scopus 로고
    • A method to avoid diverging components in the Candecomp/Parafac model for generic IxJx2 arrays
    • A. STEGEMAN and L. De LATHAUWER, "A method to avoid diverging components in the Candecomp/Parafac model for generic IxJx2 arrays," SIAM J. Matrix Ana. Appl., vol. 30, pp. 1614-1638, 2009.
    • (2009) SIAM J. Matrix Ana. Appl , vol.30 , pp. 1614-1638
    • STEGEMAN, A.1    De LATHAUWER, L.2
  • 42
    • 17444379662 scopus 로고    scopus 로고
    • Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition
    • L. De LATHAUWER,B. De MOOR, and J. VANDEWALLE, "Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition," SIAM Jour. Matrix Ana. Appl., vol. 26, no. 2, pp. 295-327, 2004.
    • (2004) SIAM Jour. Matrix Ana. Appl , vol.26 , Issue.2 , pp. 295-327
    • De LATHAUWER, L.1    De MOOR, B.2    VANDEWALLE, J.3
  • 44
    • 0039957423 scopus 로고
    • Rank and optimal computation of generic tensors
    • July
    • V. STRASSEN, "Rank and optimal computation of generic tensors," Linear Algebra Appl., vol. 52, pp. 645-685, July 1983.
    • (1983) Linear Algebra Appl , vol.52 , pp. 645-685
    • STRASSEN, V.1
  • 47
    • 33845525246 scopus 로고    scopus 로고
    • Algorithm 862: Matlab tensor classes for fast algorithm prototyping
    • B. W. BADER and T. KOLDA, "Algorithm 862: Matlab tensor classes for fast algorithm prototyping," ACM Trans. Math. Soft., vol. 32, pp. 635-653, 2006.
    • (2006) ACM Trans. Math. Soft , vol.32 , pp. 635-653
    • BADER, B.W.1    KOLDA, T.2
  • 48
    • 31044456392 scopus 로고    scopus 로고
    • M. MORUP and L. K. HANSEN et alterae, Parallel factor analysis as an exploratory tool for wavelet transformed event-related EEG, NeuroImage, 29, pp. 938-947, 2006.
    • M. MORUP and L. K. HANSEN et alterae, "Parallel factor analysis as an exploratory tool for wavelet transformed event-related EEG," NeuroImage, vol. 29, pp. 938-947, 2006.
  • 49
    • 34250499792 scopus 로고
    • Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition
    • Sept
    • J. D. CARROLL and J. J. CHANG, "Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition," Psychometrika, vol. 35, no. 3, pp. 283-319, Sept. 1970.
    • (1970) Psychometrika , vol.35 , Issue.3 , pp. 283-319
    • CARROLL, J.D.1    CHANG, J.J.2
  • 51
    • 0034861343 scopus 로고    scopus 로고
    • Positive tensor factorization
    • M. WELLING and M. WEBER, "Positive tensor factorization," Pattern Recognition Letters, vol. 22, no. 12, pp. 1255-1261, 2001.
    • (2001) Pattern Recognition Letters , vol.22 , Issue.12 , pp. 1255-1261
    • WELLING, M.1    WEBER, M.2
  • 52
    • 0036612776 scopus 로고    scopus 로고
    • Conditions for nonnegative independent component analysis
    • June
    • M. PLUMBLEY, "Conditions for nonnegative independent component analysis," Sig. Proc. Letters, vol. 9, pp. 177-180, June 2002.
    • (2002) Sig. Proc. Letters , vol.9 , pp. 177-180
    • PLUMBLEY, M.1
  • 53
    • 0033592606 scopus 로고    scopus 로고
    • Learning the parts of objects by non-negative matrix factorization
    • Oct
    • D. D. LEE and H. S. SEUNG, "Learning the parts of objects by non-negative matrix factorization.," Nature, vol. 401, pp. 788-791, Oct. 1999.
    • (1999) Nature , vol.401 , pp. 788-791
    • LEE, D.D.1    SEUNG, H.S.2
  • 54
    • 0034250083 scopus 로고    scopus 로고
    • Parallel factor analysis in sensor array processing
    • Aug
    • N. D. SIDIROPOULOS, R. BRO, and G. B. GIANNAKIS, "Parallel factor analysis in sensor array processing," IEEE Trans. Sig. Proc., vol. 48, no. 8, pp. 2377-2388, Aug. 2000.
    • (2000) IEEE Trans. Sig. Proc , vol.48 , Issue.8 , pp. 2377-2388
    • SIDIROPOULOS, N.D.1    BRO, R.2    GIANNAKIS, G.B.3
  • 55
    • 0033901839 scopus 로고    scopus 로고
    • Blind PARAFAC receivers for DS-CDMA systems
    • Mar
    • N. D. SIDIROPOULOS, G. B. GIANNAKIS, and R. BRO, "Blind PARAFAC receivers for DS-CDMA systems," IEEE Trans. on Sig. Proc., vol. 48, no. 3, pp. 810-823, Mar. 2000.
    • (2000) IEEE Trans. on Sig. Proc , vol.48 , Issue.3 , pp. 810-823
    • SIDIROPOULOS, N.D.1    GIANNAKIS, G.B.2    BRO, R.3
  • 56
    • 33745916802 scopus 로고    scopus 로고
    • Blind identification of under-determined mixtures based on the characteristic function
    • Sept
    • P. COMON and M. RAJIH, "Blind identification of under-determined mixtures based on the characteristic function," Signal Processing, vol. 86, no. 9, pp. 2271-2281, Sept. 2006.
    • (2006) Signal Processing , vol.86 , Issue.9 , pp. 2271-2281
    • COMON, P.1    RAJIH, M.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.