Mathematics in Independent Component Analysis

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78 Chapter 2. Neural Computation 16:1827-1850, 2004 A New Concept for Separability Problems in BSS 1849 The algorithm has been specified using only an energy function; gradient and fixed-point algorithms can be derived in the usual manner. Separability in nonlinear situations has turned out to be a hard problem— illposed in the most general case (Hyvärinen & Pajunen, 1999)—and not many nontrivial results exist for restricted models (Hyvärinen & Pajunen, 1999; Babaie-Zadeh et al., 2002), all only two-dimensional. We believe that this is due to the fact that the rather nontrivial proof of the Darmois- Skitovitch theorem is not at all easily generalized to more general settings (Kagan, 1986). By introducing separated functions, we are able to give a much easier proof for linear separability and also provide new results in nonlinear settings. We hope that these ideas will be used to show separability in other situations as well. Acknowledgments I thank the anonymous reviewers for their valuable suggestions, which improved the original manuscript. I also thank Peter Gruber, Wolfgang Hackenbroch, and Michaela Theis for suggestions and remarks on various aspects of the separability proof. The work described here was supported by the DFG in the grant “Nonlinearity and Nonequilibrium in Condensed Matter” and the BMBF in the ModKog project. References Babaie-Zadeh, M., Jutten, C., & Nayebi, K. (2002). A geometric approach for separating post non-linear mixtures. In Proc. of EUSIPCO ’02 (Volume 2, pp. 11–14). Toulouse, France. Bauer, H. (1996). Probability theory. Berlin: Walter de Gruyter. Bell, A., & Sejnowski, T. (1995). An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7:1129–1159. Belouchrani, A., Meraim, K. A., Cardoso, J.-F., & Moulines, E. (1997). A blind source separation technique based on second order statistics. IEEE Transactions on Signal Processing, 45(2), 434–444. Cardoso, J.-F., & Souloumiac, A. (1993). Blind beamforming for non gaussian signals. IEEE Proceedings–F, 140(6), 362–370. Cichocki, A., & Amari, S. (2002). Adaptive blind signal and image processing. New York: Wiley. Comon, P. (1994). Independent component analysis—a new concept? Signal Processing, 36:287–314. Darmois, G. (1953). Analyse générale des liaisons stochastiques. Rev. Inst. Internationale Statist., 21, 2–8. Eriksson, J., & Koivunen, V. (2003). Identifiability and separability of linear ica models revisited. In Proc. of ICA 2003, (pp. 23–27). Nara, Japan. Habl, M., Bauer, C., Puntonet, C., Rodriguez-Alvarez, M., & Lang, E. (2001). Analyzing biomedical signals with probabilistic ICA and kernel-based source
Chapter 2. Neural Computation 16:1827-1850, 2004 79 1850 F. Theis density estimation. In M. Sebaaly, (Ed.) Information science innovations (Proc.ISI’2001) (pp. 219–225). Alberta, Canada: ICSC Academic Press. Hérault, J., & Jutten, C. (1986). Space or time adaptive signal processing by neural network models. In J. Denker (Ed.), Neural networks for computing: Proceedings of the AIP Conference (pp. 206–211). New York: American Institute of Physics. Hyvärinen, A., Karhunen, J., & Oja, E. (2001). Independent component analysis. New York: Wiley. Hyvärinen, A., & Oja, E. (1997). A fast fixed-point algorithm for independent component analysis. Neural Computation, 9:1483–1492. Hyvärinen, A., & Pajunen, P. (1999). Nonlinear independent component analysis: Existence and uniqueness results. Neural Networks, 12(3), 429–439. Kagan, A. (1986). New classes of dependent random variables and a generalization of the Darmois-Skitovitch theorem to several forms. Theory Probab. Appl., 33(2), 286–295. Lee, T., & Lewicki, M. (2000). The generalized gaussian mixture model using ICA. In Proc. of ICA 2000 (pp. 239–244). Helsinki, Finland. Lin, J. (1998). Factorizing multivariate function classes. In M. Kearns, M. Jordan, & S. Solla (Eds.), Advances in neural information processing systems, 10, (pp. 563– 569). Cambridge, MA: MIT Press. Skitovitch, V. (1953). On a property of the normal distribution. DAN SSSR, 89, 217–219. Taleb, A., & Jutten, C. (1999). Source separation in post non linear mixtures. IEEE Trans. on Signal Processing, 47, 2807–2820. Theis, F., & Gruber, P. (forthcoming). Separability of analytic postnonlinear blind source separation with bounded sources. In Proc. of ESANN 2004. Evere, Belgium: d-side. Theis, F., Jung, A., Puntonet, C., & Lang, E. (2002). Linear geometric ICA: Fundamentals and algorithms. Neural Computation, 15, 1–21. Theis, F., Puntonet, C., & Lang, E. (2003). Nonlinear geometric ICA. In Proc. of ICA 2003 (pp. 275–280). Nara, Japan. Tong, L., Liu, R.-W., Soon, V., & Huang, Y.-F. (1991). Indeterminacy and identifiability of blind identification. IEEE Transactions on Circuits and Systems, 38, 499–509. Yeredor, A. (2000). Blind source separation via the second characteristic function. Signal Processing, 80(5), 897–902. Received June 27, 2003; accepted March 8, 2004.
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