Matrix Factorizations based on induced norms
Abstract
We decompose a matrix Y into a sum of bilinear terms in a stepwise manner, by considering Y as a mapping between two finite dimensional Banach spaces. We provide transition formulas, and represent them in a duality diagram, thus generalizing the well known duality diagram in the french school of data analysis. As an application, we introduce a family of Euclidean multidimensional scaling models.References
Alon, N. and Naor, A. Approximating the cut-norm via Grothendieck’s inequality. SIAM Journal on Computing, 35, 787-803, (2006).
Benzécri, J.P. (1973a). L’Analyse des Données: Vol. 2: L’Analyse des Correspondances. Paris: Dunod.
Benzécri, J.P. (1973b). L’Analyse des Données: Vol. 1: La Taxinomie. Paris: Dunod.
Benzécri, J.P. (1977). Histoire et Préhistoire de l’Analyse des Données: V: L’Analyse des correspondances. Les Cahiers de l’Analyse des Données, II(1), 9-53.
Blei, R.C. (1987). An elementary proof of the Grothendieck inequality. Proceedings of the American Mathematical Society, 100(1), 58-60.
Borg, I. and Groenen, P.G. (2005). Modern Multidimensional Scaling. 2nd edition, NY: Springer Verlag.
Boyd, D.W. (1974). The power method for lp norms. Linear Algebra and its Applications, 9, 95-101.
Burt, C. (1917). The Distribution and Relations of Educational Abilities. London, U.K: P.S. King & son.
Choulakian, V. (2003). The optimality of the centroid method. Psychometrika, 68, 473-475.
Choulakian, V. (2004). A comparison of two methods of principal component analysis. In COMPSTAT’2004 edited by J. Antoch,Physica-Verlag/Springer, 793-798.
Choulakian, V. (2005a). Transposition invariant principal component analysis in L1for long tailed data. Statistics and Probability Letters,71, 23-31.
Choulakian, V. (2005b). L1-norm projection pursuit based principal component analysis. Computational Statistics and Data Analysis, 50,1441-1451.
Choulakian, V. (2006a). Taxicab correspondence analysis. Psychometrika,71, 333-345.
Choulakian, V. (2006b). L1norm projection pursuit principal componentanalysis. Computational Statistics and Data Analysis, 50, 1441-1451.
Choulakian, V. (2008a). Taxicab correspondence analysis of contingency tables with one heavyweight column. Psychometrika, 73, 309-319.
Choulakian, V. (2008b). Multiple taxicab correspondence analysis. Advances in data Analysis and Classification, 2, 177-206.
Choulakian, V. (2010). Some numerical results on the rank of genericthree-way arrays over R. SIAM Journal of Matrix Analysis and Applications, 31(4), 1541-1551.
Choulakian, V. (2012). Picture of all solutions of successive 2-block MAXBET problems. Psychometrika, 76(4), 550-563.
Choulakian V. (2013). The simple sum score statistic in taxicab correspondence analysis. In Advances in Latent Variables (ebook), eds. Brentari E.and Carpita M., Vita e Pensiero, Milan, Italy, ISBN 978 88 343 2556 8, 6 pages.
Choulakian, V. (2014). Taxicab correspondence analysis of ratings and rankings. Journal de la Société Francaise de Statistique.155(4), 1-23.
Choulakian, V., Allard, J. and Simonetti, B. (2013). Multiple taxicab correspondence analysis of a survey related to health services. Journal of Data Science, 11(2), 205-229.
Choulakian, V. and de Tibeiro, J. (2013). Graph partitioning by correspondence analysis and taxicab correspondence analysis. Journal of Classification, 30(3), 397-427.
Choulakian, V., Simonetti, B. and Gia, T.P. (2014). Some new aspects of taxicab correspondence analysis. Statistical Methods and Applications,23(3), 401-416.
Choulakian, V., Kasparian, S., Miyake, M., Akama, H., Makoshi, N., Nakagawa, M. (2006). A statistical analysis of synoptic gospels. JADT’2006, pp.281-288.
Chu, M.T. and Funderlic, R.E. (2002). The centroid decomposition: relationships between discrete variational decompositions and SVDs. SIAM J. Matrix Analysis and Applications, 23, 1025-1044.
Chu, M.T., Funderlic, R.E. and Golub, G.H. (1995). A rank-one reduction formula and its applications to matrix factorizations. SIAM Review, 37(4), 512-530.
De La Cruz, O. and Holmes, S. (2011). The duality diagram in data analysis:Examples of modern applications. Annals of Applied Statistics, 5(4),2266-2277.
Drakakis. K. and Pearlmutter, B.A. (2009). On the calculation of the l2→ l1induced matrix norm. International Journal of Algebra, 3(5), 231-240.
Gabriel, K.R. and Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21, 489-498.
Galpin, J.S. and Hawkins, D.M. (1987). L1 estimation of a covariance matrix. Computational Statistics and Data Analysis, 5, 305-319.
Gauthier, S.M. and Choulakian, V. (2015). Taxicab correspondence analysis of abundance data in archeology: three case studies revisited. To appear in Archeologia e Calcolatori, 26.
Golub, G.H. and Van Loan, C.F. (1996). Matrix Computations. 3rd edition, John Hopkins Studies in the Mathematical Sciences, The Johns Hopkins University Press, Baltimore, MD, ISBN 978-0-8018-5414-9.
Gower, J.C. (1966). Some distance properties of latent root and vector methods in multivariate analysis. Biometrika, 53, 325-338.
Guttman, L. (1944). General theory and methods for matric factoring. Psychometrika, 9, 1-16.
Harman, H.H. (1967). Modern Factor Analysis. Chicago, IL: The University of Chicago Press.
Holmes, S. (2008). Mutivariate analysis: The French way. In Probability and Statistics: Essays in honor of David Freedman, eds. D. Nolan and T. Speed, pp. 219-233. Ohio, USA: Institute of Mathematical Statistics.
Horn, R.A. and Johnson, C.R. (1990). Matrix Analysis. Cambridge University Press, NY.
Horst, P. (1965). Factor Analysis of Data Matrices. NY: Holt, Rinehart and Winston.
Hubert, L., Meulman, J. and Heiser, W. (2000). Two purposes for matrix factorization: A historical appraisal. SIAM Review, 42, 68-82.
Jameson, G.J.O. (1987). Summing and Nuclear Norms in Banach Space Theory. London Mathematical Society Student Texts 8, Cambridge University Press, Cambridge.
Kreyszig, E. (1978). Introductory Functional Analysis with Applications J. Wiley and Sons, N.Y.
Kwak, N. (2008). Principal component analysis based on L1-norm maximization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(9), 1672-1680.
Lewis, A.D. (2010). A top nine list: Most popular induced matrix norms. Downloaded from http://www.mast.queensu.ca/~andrew/notes/pdf/2010b.pdf
McCoy, M. and Tropp, J. (2011). Two proposals for robust PCA usingsemidefinite programming. Electronic Journal of Statistics, 5,1123-1160.
Mulaik, S. (1987). A brief history of the phlosophical foundations of exploratory factor analysis. Multivariate Behavioral Research, 22, 267-305.
Pisier, G. (2012). Grothendieck’s theorem, past and present. Bulletin of the American Mathematical Society, 49 (2): 237–323.
Rohn, J. (2000). Computing the norm ∥A∥∞−→1 is NP-hard. Linear and Multilinear Algebra, 47 (3), 195-204.
Rhodes, J.A. (2010). A concise proof of Kruskal’s theorem on tensor decomposition. Linear Algebra and its Applications, 432 (7), 1818-1824.
Ten Berge, J.M.F. (2011). Simplicity and typical rank results for three-way arrays. Psychometrika, 76(1), 3-12.
Thurstone, L.L. (1931). Multiple factor analysis. Psychological Review, 38, 406-427.
Thurstone, L.L. (1947). Multiple-Factor Analysis. The University of Chicago Press, Chicago.
Torgerson, W.S. (1952). Multidimensional scaling: 1. Theory and method. Psychometrika, 17, 401-419.
Wold, H. (1966). Estimation of principal components and related models by iterative least squares. In Multivariate Analysis, ed. Krishnaiah,P.R., N.Y: Academic Press, 391-420.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
