Abstract: Fast Independent Component Analysis（FastICA） is the most widely used method to solve blind source separation problems. Since only limited data samples can be obtained in practice, the FastICA algorithm usually used is based on sample. However, the common convergence analysis of the FastICA algorithm is with respect to the ensemble FastICA algorithm. Therefore, it is of great significance to explore convergence and consistency of the FastICA algorithm based on sample. Firstly, the relevant convergence properties of the ensemble FastICA algorithm are proved in a more concise way, including the relationship between the local maximum of the contrast function and the fixed point of the FastICA iterative function. Secondly, the Dirac function is introduced to construct the probability density function of observed signals. The convergence condition of the FastICA algorithm based on sample is given by the law of large numbers. Finally, based on the M-estimator consistency theorem, it is proved that the FastICA estimator is a consistent estimator. The results of the simulation experiments verify the consistency of the FastICA estimator.

Key words: Fast Independent Component Analysis（FastICA）, convergence, fixed point, consistent converge in probability, consistency

摘要： 快速独立成分分析（Fast Independent Component Analysis，FastICA）是解决盲源分离问题使用最广泛的方法。在实际中，只能得到有限数据样本，所以采用的均是基于样本的FastICA算法。而常见的FastICA算法的收敛性分析均属于全集FastICA算法的收敛性分析，所以研究基于样本FastICA算法的收敛性和算法的一致性有至关重要的意义。以一种更简洁的方法证明了全集FastICA的相关收敛性质，包括对比函数的局部极大值和FastICA迭代函数不动点之间的关系。引入狄拉克函数，构造观测信号的概率密度函数，通过大数定律，给出了基于样本的FastICA算法收敛性条件。依据M-估计一致性定理，证明了FastICA给出的估计是一致估计。仿真实验的结果验证了FastICA估计的一致性。

关键词: 快速独立成分分析（FastICA）, 收敛性, 不动点, 依概率一致收敛, 一致性