0, K.i > 0, K.iK.ja;aJ > 1, \n\n(22) \n\nfor i , j = 1,\" ' , n, where mi = E(c.p'(Yi (k))y;(k)], K.i = E[c.p~(Yi)], a; = E[IYiI 2 ]. \nTherefore, we have the following theorem: \n\nTheorem 2 If the conditions (22) are satisfied, then the natural gradient learning algo(cid:173)\nrithm (20) is locally stable. \n\n\fBlind Deconvolution of Nonminimum Phase Systems \n\n369 \n\nReferences \n\n[1] S. Amari. Differential-geometrical methods in statistics, Lecture Notes in Statistics, \n\nvolume 28. Springer, Berlin, 1985. \n\n[2] S. Amari. Natural gradient works efficiently in learning. Neural Computation, \n\n10:251-276, 1998. \n\n[3 J S. Amari. ICA of temporally correlated signals - Learning algorithm. In Proceeding \nof 1st Inter. Workshop on Independent Component Analysis and Signal Separation, \npages 37-42, Aussois, France, January, 11-15 1999. \n\n[4] S. Amari. Superefficiency in blind source separation. IEEE Trans. on Signal Process(cid:173)\n\ning, 47(4):936-944, April 1999. \n\n[5] S. Amari and J.-F. Cardoso. Blind source separation- semiparametric statistical ap(cid:173)\n\nproach. IEEE Trans. Signal Processing, 45:2692-2700, Nov. 1997. \n\n[6] S. Amari, T. Chen, and A. Cichocki. Nonholonomic orthogonal constraints in blind \n\nsource separation. Neural Comput., to be published. \n\n[7] S. Amari and A. Cichocki. Adaptive blind signal processing- neural network ap(cid:173)\n\nproaches. Proceedings of the IEEE, 86(10):2026-2048,1998. \n\n[81 S. Amari, S. Douglas, A. Cichocki, and H. Yang. Multichannel blind deconvolution \n\nand equalization using the natural gradient. In Proc. IEEE Workshop on Signal Pro(cid:173)\ncessing Adv. in Wireless Communications, pages 101-104, Paris, France, April 1997. \n[9] S. Amari and M. Kawanabe. Estimating functions in semiparametric statistical mod(cid:173)\nels. In I. V. Basawa, v.P. Godambe, and R.L. Taylor, editors, Estimating Functions, \nvolume 32 of Monograph Series, pages 65-81. IMS, 1998. \n\n[10] S. Amari and M. Kumon. Estimation in the presence of infinitely many nuisance \nparameters in semiparametric statistical models. Ann. Statistics, 16: 1044-1068, 1988. \n\n[11] A.l. Bell and T.l. Sejnowski. An information maximization approach to blind sepa(cid:173)\n\nration and blind deconvolution. Neural Computation, 7: 1129-1159, 1995. \n\n[12] P. Bickel, C. Klaassen, Y. Ritov, and J. Wellner. Efficient and Adaptive Estimation \nfor Semiparametric Models. The Johns Hopkins Univ. Press, Baltimore and London, \n1993. \n\n[13] J.-F Cardoso. Blind signal separation: Statistical principles. Proceedings of the IEEE, \n\n86(10):2009-2025,1998. \n\n[14] J.-F. Cardoso and B. Laheld. Equivariant adaptive source separation. IEEE Trans. \n\nSignal Processing, SP-43: 30 17-3029, Dec 1996. \n\n[15] A. Cichocki and R. Unbehauen. Robust neural networks with on-line learning for \nblind identification and blind separation of sources. IEEE Trans Circuits and Systems \nI: Fundamentals Theory and Applications, 43(11):894-906, 1996. \n\n[16] L. Tong, R.W. Liu, v.c. Soon, and Y.F. Huang. Indeterminacy and identifiability of \n\nblind identification. IEEE Trans. Circuits, Syst., 38(5):499-509, May 1991. \n\n[17] H. Yang and S. Amari. Adaptive on-line learning algorithms for blind separation: \nMaximum entropy and minimal mutual infonnation. Neural Comput., 9: 1457-1482, \n1997. \n\n[18] L. Zhang, A. Cichocki, and S. Amari. Geometrical structures of FIR manifold and \ntheir application to multichannel blind deconvolution. In Proceeding of NNSP'99, \npages 303-312, Madison, Wisconsin, August 23-25 1999. \n\n[19] L. Zhang, A. Cichocki, and S. Amari. Multichannel blind deconvolution of non(cid:173)\n\nminimum phase systems using information backpropagation. In Proceedings of the \nFifth International Conference on Neural Information Processing(ICONIP'99), page \n210-216, Perth, Australia, Nov. 16-20 1999. \n\n\f", "award": [], "sourceid": 1653, "authors": [{"given_name": "Liqing", "family_name": "Zhang", "institution": null}, {"given_name": "Shun-ichi", "family_name": "Amari", "institution": null}, {"given_name": "Andrzej", "family_name": "Cichocki", "institution": null}]}