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Mathematics in Independent Component Analysis

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Chapter 12. LNCS 3195:718-725, 2004 183<br />

8 Fabian J. Theis and Shun-ichi Amari<br />

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(a) sources<br />

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(c) mixture scatterplot<br />

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(b) mixtures<br />

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(d) recovered sources<br />

Fig. 2. Example: (a) shows the 1-sparse source signals, and (b) the postnonl<strong>in</strong>ear overcomplete<br />

mixtures. The orig<strong>in</strong>al source directions can be clearly seen <strong>in</strong> the structure<br />

of the mixture scatterplot (c). The crosses and stars <strong>in</strong>dicate the found <strong>in</strong>terpolation<br />

po<strong>in</strong>ts used for approximat<strong>in</strong>g the separat<strong>in</strong>g nonl<strong>in</strong>earities, generated by geometrical<br />

preprocess<strong>in</strong>g. Now, accord<strong>in</strong>g to theorem 3, the sources can be recovered uniquely,<br />

figure (d), except for permutation and scal<strong>in</strong>g.<br />

7. Chen, S., Donoho, D., Saunders, M.: Atomic decomposition by basis pursuit. SIAM<br />

J. Sci. Comput. 20 (1998) 33–61<br />

8. Georgiev, P., Theis, F., Cichocki, A.: Bl<strong>in</strong>d source separation and sparse component<br />

analysis of overcomplete mixtures. In: Proc. of ICASSP 2004, Montreal, Canada<br />

(2004)<br />

9. Taleb, A., Jutten, C.: Indeterm<strong>in</strong>acy and identifiability of bl<strong>in</strong>d identification.<br />

IEEE Transactions on Signal Process<strong>in</strong>g 47 (1999) 2807–2820<br />

10. Babaie-Zadeh, M., Jutten, C., Nayebi, K.: A geometric approach for separat<strong>in</strong>g<br />

post non-l<strong>in</strong>ear mixtures. In: Proc. of EUSIPCO ’02. Volume II., Toulouse, France<br />

(2002) 11–14<br />

11. Amari, S., Park, H., Fukumizu, K.: Adaptive method of realiz<strong>in</strong>g gradient learn<strong>in</strong>g<br />

for multilayer perceptrons. Neural Computation 12 (2000) 1399–1409

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