Abstract
Independent component analysis (ICA) neural networks can estimate independent components from the mixed signal. The dynamical behavior of the learning algorithms for ICA neural networks is crucial to effectively apply these networks to practical applications. The paper presents the stability and chaotic dynamical behavior of a class of ICA learning algorithms with constant learning rates. Some invariant sets are obtained so that the non-divergence of these algorithms can be guaranteed. In these invariant sets, the stability and chaotic behaviors are analyzed. The conditions for stability and chaos are derived. Lyapunov exponents and bifurcation diagrams are presented to illustrate the existence of chaotic behavior.
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References
Comon P (1994) Independent component analysis - a new concept. Signal Process 36: 287–314
Jutten C, Herault J (1991) Blind separation of sources part I an adaptive algorithm based on neuromimetic architecture. Signal Process 24: 1–10
Bell A, Sejnowski TJ (1997) Edges are the independent components of natural scenes. In: Advances in neural information processing 9 (NIPS 96). MIT Press, pp 831–837
Karhunen J, Hyvärinen A, Vigario R, Hurri J, Oja E (1997) Applications of neural blind separation to signal and image processing. In: Proceedings of IEEE international conference on acoustics, speech and signal processing (ICASSP 97). Munich, Germany, pp 131–134
Kotani M, Ozawa S (2005) Feature extraction using independent components of each category. Neural Process Lett 22: 113–124
Donoho D (1981) On minimum entropy deconvolution. In: Applied Time Series Analysis II. Academic Press, pp 565–608
Cichocki A, Thawonmas R (2000) On-line algorithms for blind signal extraction of arbitrarily distributed, but temporally correlated sources using second order statistics. Neural Process Lett 12: 91–98
Hyvärinen A, Oja E (1997) Simple neuron models for independent component analysis. Int J Neural Syst 7(6): 671–687
Hyvärinen A, Oja E (1998) Independent component analysis by general non-linear hebbian-like learning Rues. Signal Process 64(3): 301–313
Ilin A, Valpola H (2005) On the effect of the form of the posterior approximation in variational learning of ICA models. Neural Process Lett 22: 183–204
Karhunen J, Oja E, Wang L, Vigario R, Joutsensalo J (1997) A class of neural networks for independent component analysis. IEEE Trans Neural Netw 8(3): 486–504
Karhunen J (2005) A resampling test for the total independence of stationary time series: application to the performance evaluation of ICA algorithms. Neural Process Lett 22: 311–324
Mollah MNH, Eguchi S, Minami M (2007) Robust prewhitening for ICA by minimizing β-divergence and its application to fastICA. Neural Process lett 25: 91–110
Oja E, Yuan Z (2006) The fastICA algorithm revisited: convergence analysis. IEEE Trans Neural Netw 17(6): 1370–1381
Wang X, Tian L (2006) Bifurcation analysis and linear control of the Newton-Leipnik system. Chaos Solitons Fractals 27: 31–38
AGIZA HN (1999) On the analysis of stability, bifurcation, chaos and chaos control of Kopel Map. Chaos Solitons Fractals 10: 1909–1916
Cheng Z, Lin Y, Cao J (2006) Dynamical behaviors of a partial-dependent predator–prey system. Chaos Solitons Fractals 28: 67–75
Jing Z, Yang J (2006) Bifurcation and chaos in discrete-time predator–prey system. Chaos Solitons Fractals 27: 259–277
Li C, Chen G (2004) Chaos in the fractional order Chen system and its control. Chaos Solitons Fractals 22: 549–554
Lv JC, Yi Z (2007) Stability and chaos of LMSER PCA learning algorithm. Chaos Solitons Fractals 32: 1440–1447
Lv JC, Yi Z, Tan KK (2007) Global convergence of GHA learning algorithm with nonzero-approaching learning rates. IEEE Trans Neural Netw 18(6): 1557–1571
Lv JC, Yi Z, Tan KK (2006) Global convergence of Oja’s PCA learning algorithm with a non-zero-approaching adaptive learning rate. Theor Comput Sci 367: 286–307
Lv JC, Yi Z, Tan KK (2006) Convergence analysis of Xu’s LMSER learning algorithm via deterministic discrete time system method. Neurocomputing 70: 362–372
Xu L, Oja E, Suen CY (1992) Modified hebbian learning for curve and surface fitting. Neural Netw 5: 441–457
Yi Z, Ye M, Lv JC, Tan KK (2005) Convergence analysis of a deterministic discrete time system of Oja’s PCA learning algorithm. IEEE Trans Neural Netw 16(6): 1318–1328
Zufiria PJ (2002) On the discrete-time dynamic of the basic Hebbian neural-network nodes. IEEE Trans Neural Netw 13(6): 1342–1352
Yi Z, Tan KK (2004) Convergence analysis of recurrent neural networks. Kluwer Academic Publishers
Dror G, Tsodyks M (2000) Chaos in neural networks with dynamical synapses. Neurocomputing 32: 365–370
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Lv, J.C., Tan, K.K., Yi, Z. et al. Stability and Chaos of a Class of Learning Algorithms for ICA Neural Networks. Neural Process Lett 28, 35–47 (2008). https://doi.org/10.1007/s11063-008-9080-2
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DOI: https://doi.org/10.1007/s11063-008-9080-2