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A Further Result on the ICA One-Bit-Matching Conjecture

jwma{at}math.pku.edu.cn, Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, N.T., Hong Kong, and School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, China

Zhiyong Liu

zyliu{at}cuhk.edu.hk, Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Lei Xu

lxu{at}cse.cuhk.edu.hk, Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, N.T., Hong Kong

The one-bit-matching conjecture for independent component analysis (ICA) has been widely believed in the ICA community. Theoretically, it has been proved that under the assumption of zero skewness for the model probability density functions, the global maximum of a cost function derived from the typical objective function on the ICA problem with the one-bit-matching condition corresponds to a feasible solution of the ICA problem. In this note, we further prove that all the local maximums of the cost function correspond to the feasible solutions of the ICA problem in the two-source case under the same assumption. That is, as long as the one-bit-matching condition is satisfied, the two-source ICA problem can be successfully solved using any local descent algorithm of the typical objective function with the assumption of zero skewness for all the model probability density functions.