带参考信号的独立分量分析理论及其应用研究
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摘要
对多元统计信号的探索一直是信息科学领域的热门研究课题,如何发现多元信号中的内在因素或者分量是该领域的研究重点之一。独立分量分析这种多元统计分析方法作为近年来人工神经网络、统计学习理论、信号处理等研究领域中的热点问题,它的一些成果已经应用到多个实际问题当中,证明了其广阔的应用价值。
独立分量分析是盲源信号处理的一个分支,该理论假设多元观察信号是由多个相互独立的非高斯信号线性混合而成的。在很多实际应用领域中,我们对所需要获得的独立分量有部分先验信息,然而经典的独立分量分析方法首先需要把所有的独立分量都恢复出来,然后再由用户通过先验信息来选择自己所感兴趣的独立分量。这样的方法一般存在耗时,输出结果不稳定等问题。在遇到高维问题时,经典方法甚至可能出现完全失效的情况。本文研究的核心内容就是如何使用带先验信息的参考信号来直接抽取所感兴趣的独立分量。
本文的主要贡献总结为如下几点：
1.严格推导了约束独立分量分析框架下的带参考信号的独立分量分析问题的学习公式,发现原来的算法中存在的推导错误等,提出了修正的约束独立分量分析框架下的带参考信号的独立分量分析算法,并在理论上证明了算法改进的正确性,同时通过模拟实验和实际数据实验验证了修正后算法的有效性。
2.研究了约束独立分量分析框架下的带参考信号的独立分量分析的收敛不稳定问题,发现了问题的原因在于ICA-R优化目标函数在不等式约的可行区域内不一定为凸函数,也就是KT条件不是全局充分的,那么采用类似牛顿法的优化算法是不能够保证获得全局最优值的。一般来说,采用贪婪策略的算法(每一次迭代都降低目标函数)都会出现误收敛的情况,所以算法的输出不稳定。在这种情况下,我们提出了新的ICA-R算法,在算法中我们引入了在算法早期判断误收敛的技术,解决了算法误收敛的问题,并且在实验中证明了新算法的误收敛率为0。
3.研究了ICA-R的应用方法。在ICA-R应用中距离阈值参数的选择往往决定了算法的成功和失败。我们提出了合理选择ICA-R阈值参数方法,包括参考信号收缩方法来灵活的选择参数等。我们提出并证明了直接采用观测通道作为参考信号的可行性,简化了ICA-R的应用,使得参考信号在某些条件下并不需要手工构造就可以直接获得。
4.提出了Complete ICA-R的方法。通过研究发现Complete ICA-R和传统ICA方法相比,是一种非常灵活的ICA算法,它不再通过强行的去相关化来防止同一性收敛问题,因此它的输出更好的保留了数据的内部特性。我们通过理论证明和实验结果说明了Complete ICA-R的输出质量确实要优于传统的ICA算法,特别是Complete ICA-R在某些应用中(观测信号多于独立分量时)还可以帮助研究人员判断独立分量的数量。
5.提出了非约束独立分量分析框架下的带参考信号的独立分量分析方法。我们认为作为先验信息载体的参考信号不仅可以用作ICA算法的约束条件,从而提出约束独立分量分析框架,而且也可以用在ICA算法之前使用参考信号。我们提出了采用参考信号来设置初始权值向量的方法,发现使用这个预设的权值向量后,ICA算法可以收敛到需要的独立分量上,以完成带参考信号独立分量分析的任务。并且我们证明了这个方法的有效性和适用条件。
6.发现并证明了原来约束独立分量分析框架的理论漏洞,提出了新的普适的带参考信号的独立分量分析算法框架,弥补了旧理论框架的漏洞。
The study of multivariate stochastic signals is a hot topic in information science field. The mainstream researches on this topic are how to discover the inherent factors or components from the multivariate stochastic signals. Recently, the independent component analysis (ICA) becomes a hotspot problem in artificial neural networks, statistical learning, signal processing, and etc. The high applied value of ICA has been verified by many successful applications of ICA in various fields.
ICA is a research branch of blind source separation (BSS), which theoretically assumes that the observed multivariate signals are the linear mixtures of independent components (ICs) with non-Gaussian distributions. In many fields, researchers have some prior information on the desired ICs. However, the classic ICA methods have to calculate all the ICs and then choose the desired ICs by post-selection which is not only time-consuming but also unstable. And when the dimentsion of input data is high, the classic ICA methods maybe unable toproduce any correct results. The aim of this paper is to study how to use prior information, in the form of reference signals, to extract the desired ICs directly.
The main contributions of the paper can be summarized as follows:
1. We rigidly derived algorithm for the ICA with reference signals (ICA-R) under constrained ICA (cICA) framework, and found flaws in the previous works, then proposed an improved algorithm whose validity was proved not only by theoretical analyses but also by simulation on both artificial and real-world data.
2. We probed problem of the unstable convergence of those previous ICA-R algorithms (under the framework of cICA) and found the reason was that the ICA contrast function can be nonconvex in the feasible region defined by the inequality constraint, which means that the KT condition is not globally sufficient. As the result, using Newton-like optimization algorithm (and other greedy algorithm) cannot guarantee the correct convergence. The proposed new ICA-R algorithm which can predict the future incorrect convergence was presented so as to avoid the instability of the previous algorithms which was confirmed by our experiments from which no incorrect convergence of the new algorithm was observed.
3. We studied how to facilitate the application of the ICA-R algorithm. Since the parameter (or threshold) measuring the distance between reference signal and output is critical, we proposed a method that facilitates the parameter selection including reference deflation method and etc. Then we suggested that it is convenient to use one of the observed data channel as reference signal in some applications instead of concocting reference signal by manpower.
4. We proposed the Complete ICA-R method for recovering all the underlying components as other classic ICAs done but without compulsively decorrelating the outputs. Therefore, the Complete ICA-R is flexible so as to remain the inherent structures of the outputs. The experimental results indicated that the Complete ICA-R algorithm would produce the outputs with better quality result compared to the classic methods. Particularly, the proposed Complete ICA-R has a unique feature which could help the users to judge the number of ICs when the number of the observed channels are more than the number of underlying components.
5. A new method for ICA-R was presented which was not under the framework of cICA. The prior information in the form of reference signal should be used not only as a constraint which was the basic ideal of cICA but also before the learning of the weight vector. In this paper, an approach was proposed which used the reference to build the initial weight vector. It was found that the one-unit ICA algorithm could find the right demixing vector with the predesigned weight vector under some conditions. Moreover, the validity of the approach was proved and the detailed algorithm was presented.
6. The theoretical flaw of the framework of cICA was found and proved in our work. In addition, we also proposed the new framework of ICA-R which can remedy the previous general framework of ICA-R.
独立分量分析是盲源信号处理的一个分支,该理论假设多元观察信号是由多个相互独立的非高斯信号线性混合而成的。在很多实际应用领域中,我们对所需要获得的独立分量有部分先验信息,然而经典的独立分量分析方法首先需要把所有的独立分量都恢复出来,然后再由用户通过先验信息来选择自己所感兴趣的独立分量。这样的方法一般存在耗时,输出结果不稳定等问题。在遇到高维问题时,经典方法甚至可能出现完全失效的情况。本文研究的核心内容就是如何使用带先验信息的参考信号来直接抽取所感兴趣的独立分量。
本文的主要贡献总结为如下几点：
1.严格推导了约束独立分量分析框架下的带参考信号的独立分量分析问题的学习公式,发现原来的算法中存在的推导错误等,提出了修正的约束独立分量分析框架下的带参考信号的独立分量分析算法,并在理论上证明了算法改进的正确性,同时通过模拟实验和实际数据实验验证了修正后算法的有效性。
2.研究了约束独立分量分析框架下的带参考信号的独立分量分析的收敛不稳定问题,发现了问题的原因在于ICA-R优化目标函数在不等式约的可行区域内不一定为凸函数,也就是KT条件不是全局充分的,那么采用类似牛顿法的优化算法是不能够保证获得全局最优值的。一般来说,采用贪婪策略的算法(每一次迭代都降低目标函数)都会出现误收敛的情况,所以算法的输出不稳定。在这种情况下,我们提出了新的ICA-R算法,在算法中我们引入了在算法早期判断误收敛的技术,解决了算法误收敛的问题,并且在实验中证明了新算法的误收敛率为0。
3.研究了ICA-R的应用方法。在ICA-R应用中距离阈值参数的选择往往决定了算法的成功和失败。我们提出了合理选择ICA-R阈值参数方法,包括参考信号收缩方法来灵活的选择参数等。我们提出并证明了直接采用观测通道作为参考信号的可行性,简化了ICA-R的应用,使得参考信号在某些条件下并不需要手工构造就可以直接获得。
4.提出了Complete ICA-R的方法。通过研究发现Complete ICA-R和传统ICA方法相比,是一种非常灵活的ICA算法,它不再通过强行的去相关化来防止同一性收敛问题,因此它的输出更好的保留了数据的内部特性。我们通过理论证明和实验结果说明了Complete ICA-R的输出质量确实要优于传统的ICA算法,特别是Complete ICA-R在某些应用中(观测信号多于独立分量时)还可以帮助研究人员判断独立分量的数量。
5.提出了非约束独立分量分析框架下的带参考信号的独立分量分析方法。我们认为作为先验信息载体的参考信号不仅可以用作ICA算法的约束条件,从而提出约束独立分量分析框架,而且也可以用在ICA算法之前使用参考信号。我们提出了采用参考信号来设置初始权值向量的方法,发现使用这个预设的权值向量后,ICA算法可以收敛到需要的独立分量上,以完成带参考信号独立分量分析的任务。并且我们证明了这个方法的有效性和适用条件。
6.发现并证明了原来约束独立分量分析框架的理论漏洞,提出了新的普适的带参考信号的独立分量分析算法框架,弥补了旧理论框架的漏洞。
The study of multivariate stochastic signals is a hot topic in information science field. The mainstream researches on this topic are how to discover the inherent factors or components from the multivariate stochastic signals. Recently, the independent component analysis (ICA) becomes a hotspot problem in artificial neural networks, statistical learning, signal processing, and etc. The high applied value of ICA has been verified by many successful applications of ICA in various fields.
ICA is a research branch of blind source separation (BSS), which theoretically assumes that the observed multivariate signals are the linear mixtures of independent components (ICs) with non-Gaussian distributions. In many fields, researchers have some prior information on the desired ICs. However, the classic ICA methods have to calculate all the ICs and then choose the desired ICs by post-selection which is not only time-consuming but also unstable. And when the dimentsion of input data is high, the classic ICA methods maybe unable toproduce any correct results. The aim of this paper is to study how to use prior information, in the form of reference signals, to extract the desired ICs directly.
The main contributions of the paper can be summarized as follows:
1. We rigidly derived algorithm for the ICA with reference signals (ICA-R) under constrained ICA (cICA) framework, and found flaws in the previous works, then proposed an improved algorithm whose validity was proved not only by theoretical analyses but also by simulation on both artificial and real-world data.
2. We probed problem of the unstable convergence of those previous ICA-R algorithms (under the framework of cICA) and found the reason was that the ICA contrast function can be nonconvex in the feasible region defined by the inequality constraint, which means that the KT condition is not globally sufficient. As the result, using Newton-like optimization algorithm (and other greedy algorithm) cannot guarantee the correct convergence. The proposed new ICA-R algorithm which can predict the future incorrect convergence was presented so as to avoid the instability of the previous algorithms which was confirmed by our experiments from which no incorrect convergence of the new algorithm was observed.
3. We studied how to facilitate the application of the ICA-R algorithm. Since the parameter (or threshold) measuring the distance between reference signal and output is critical, we proposed a method that facilitates the parameter selection including reference deflation method and etc. Then we suggested that it is convenient to use one of the observed data channel as reference signal in some applications instead of concocting reference signal by manpower.
4. We proposed the Complete ICA-R method for recovering all the underlying components as other classic ICAs done but without compulsively decorrelating the outputs. Therefore, the Complete ICA-R is flexible so as to remain the inherent structures of the outputs. The experimental results indicated that the Complete ICA-R algorithm would produce the outputs with better quality result compared to the classic methods. Particularly, the proposed Complete ICA-R has a unique feature which could help the users to judge the number of ICs when the number of the observed channels are more than the number of underlying components.
5. A new method for ICA-R was presented which was not under the framework of cICA. The prior information in the form of reference signal should be used not only as a constraint which was the basic ideal of cICA but also before the learning of the weight vector. In this paper, an approach was proposed which used the reference to build the initial weight vector. It was found that the one-unit ICA algorithm could find the right demixing vector with the predesigned weight vector under some conditions. Moreover, the validity of the approach was proved and the detailed algorithm was presented.
6. The theoretical flaw of the framework of cICA was found and proved in our work. In addition, we also proposed the new framework of ICA-R which can remedy the previous general framework of ICA-R.
引文
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