Mathematics in Independent Component Analysis

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112 Chapter 5. IEICE TF E87-A(9):2355-2363, 2004 THEIS and NAKAMURA: QUADRATIC INDEPENDENT COMPONENT ANALYSIS −0.17 0.16 −0.25 0.08 0.16 −0.11 0.28 −0.10 0.19 −0.13 −0.24 0.23 −0.20 0.12 −0.27 0.27 0.19 −0.16 −0.17 0.16 −0.30 0.09 0.38 0.05 −0.19 0.17 0.18 −0.12 −0.24 0.24 −0.22 0.22 −0.24 0.22 0.24 −0.17 0.24 −0.09 0.25 −0.22 0.23 −0.15 0.19 −0.17 0.18 −0.16 0.21 −0.21 0.26 −0.20 0.33 −0.15 −0.28 0.15 −0.17 0.17 −0.17 0.16 −0.23 0.23 −0.24 0.23 0.23 −0.22 0.20 −0.20 −0.30 0.30 0.18 −0.16 −0.18 0.15 0.28 −0.24 −0.23 0.18 0.20 −0.20 −0.26 0.26 0.19 −0.19 −0.22 0.22 0.31 −0.30 0.26 −0.23 0.18 −0.13 0.15 −0.15 0.18 −0.18 −0.31 0.30 −0.23 0.17 −0.25 0.24 −0.27 0.26 0.25 −0.23 −0.19 0.17 0.11 −0.11 −0.08 0.06 −0.15 0.13 0.25 −0.24 0.28 −0.26 −0.23 0.20 −0.18 0.15 −0.22 0.19 −0.22 0.18 0.11 −0.09 0.28 −0.28 Fig. 9 Quadratic ICA of natural images. 3·10 5 sample pictures of size 8 × 8 were used. Plotted are the recovered maximal filters of i.e. rows of the eigenvalue matrices of the quadratic form coefficient matrices (top figure). For each component the two largest filters (with respect to the absolute eigenvalues) are shown (altogether 2 · 64). Above each image the corresponding eigenvalue (multiplied with 10 3 ) is printed. In the bottom figure, the absolute values of the 10 largest eigenvalues of each filter are shown. Clearly, in most filters one or two eigenvalues are dominant. ’Nonlinearity and Nonequilibrium in Condensed Matter’ and the BMBF in the project ’ModKog’. References [1] K. Abed-Meraim, A. Belouchrani, and Y. Hua. Blind identification of a linear-quadratic mixture of independent components based on joint diagonalization procedure. In Proc. of ICASSP 1996, volume 5, pages 2718–2721, Atlanta, USA, 1996. [2] H. Bartsch and K. Obermayer. Second order statistics of natural images. Neurocomputing, 52-54:467–472, 2003. [3] A. Cichocki and S. Amari. Adaptive blind signal and image processing. John Wiley & Sons, 2002. [4] A. Cichocki and R. Unbehauen. Robust estimation of principal components in real time. Electronics Letters, 29(21):1869–1870, 1993. [5] P. Comon. Independent component analysis - a new concept? Signal Processing, 36:287–314, 1994. [6] G. Darmois. Analyse générale des liaisons stochastiques. Rev. Inst. Internationale Statist., 21:2–8, 1953. [7] J. Eriksson and V. Koivunen. Identifiability and separability of linear ica models revisited. In Proc. of ICA 2003, pages 23–27, 2003. [8] P.G. Georgiev. Blind source separation of bilinearly mixed signals. In Proc. of ICA 2001, pages 328–330, San Diego, USA, 2001. [9] W. Hashimoto. Quadratic forms in natural images. Network: Computation in Neural Systems, 14:765–788, 2003. [10] S. Hosseini and Y. Deville. Blind separation of linearquadratic mixtures of real sources using a recurrent structure. Lecture Notes in Computer Science, 2687:241–248, 2003. [11] A. Hyvärinen. Fast and robust fixed-point algorithms for independent component analysis. IEEE Transactions on Neural Networks, 10(3):626–634, 1999. [12] A. Hyvärinen, J.Karhunen, and E.Oja. Independent Component Analysis. John Wiley & Sons, 2001. [13] A. Hyvärinen and P. Pajunen. On existence and uniqueness of solutions in nonlinear independent component analysis. Proceedings of the 1998 IEEE International Joint Conference on Neural Networks (IJCNN’98), 2:1350–1355, 1998. [14] M. Joho, H. Mathis, and R.H. Lamber. Overdetermined blind source separation: using more sensors than source signals in a noisy mixture. In Proc. of ICA 2000, pages 81–86, Helsinki, Finland, 2000. [15] A. Leshem. Source separation using bilinear forms. In Proc. of the 8th Int. Conference on Higher-Order Statistical Signal Processing, 1999. [16] V.P. Skitovitch. On a property of the normal distribution. DAN SSSR, 89:217–219, 1953. [17] F.J. Theis. A new concept for separability problems in blind source separation. Neural Computation accepted, 2004. [18] J.H. van Hateren and D.L. Ruderman. Independent component analysis of natural image sequences yields spatiotemporal filters similar to simple cells in primary visual cortex. Proc. R. Soc. Lond. B, 265:2315–2320, 1998. [19] S. Winter, H. Sawada, and S. Makino. Geometrical interpretation of the PCA subspace method for overdetermined blind source separation. In Proc. of ICA 2003, pages 775– 780, Nara, Japan, 2003. 9
Chapter 6 LNCS 3195:726-733, 2004 Paper F.J. Theis, A. Meyer-Bäse, and E.W. Lang. Second-order blind source separation based on multi-dimensional autocovariances. In Proc. ICA 2004, volume 3195 of LNCS, pages 726-733, Granada, Spain, 2004. Springer Reference (Theis et al., 2004e) Summary in section 1.3.1 113
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Statistical machine learning of bio
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vi Preface
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viii CONTENTS 3 Signal Processing 8
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Chapter 1 Statistical machine learn
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1.1. Introduction 5 auditory cortex
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S U M M A R Y T his �le contains
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298 BIBLIOGRAPHY Barber, C., Dobkin
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300 BIBLIOGRAPHY Georgiev, P. (2001
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302 BIBLIOGRAPHY Keck, I., Theis, F
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306 BIBLIOGRAPHY Theis, F. and Inou
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