盲信号分离若干关键问题研究
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摘要
盲信号分离在众多科学领域,特别是在语音信号分离与识别、生物信号(如脑电图、心电图)处理、无线通信系统等领域,有着极其广泛的应用。实际上,由于其重要的理论价值和广泛的应用前景,盲信号分离已经成为当前信号处理领域最热门的新技术之一。经过大量学者的不懈努力,盲信号分离已经在多个方面得到深入的研究和发展,并涌现了大量优秀的盲分离算法。然而,盲分离仍然存在一些关键理论与实际问题需要解决。本博士论文针对这些问题展开研究,主要作出了如下创新性工作:
1.改进和推广了著名的盲分离几何算法——最小值域方法。盲分离几何算法能够为盲分离提供可视化的解释和分离过程。其中,最小值域方法有比较严格的理论基础,且对观测信号的个数没有限制。然而,其分离算法的可靠性和效率都存在不足。通过利用凸包的一些优秀性质,本文提出的改进算法效率更高,可靠性更好。另外,本文把最小值域算法推广到最大值域方法,从而扩展了这类几何算法的应用范围。
2.首次对对角化器的条件数展开深入研究。联合对角化是解决盲分离的最重要工具之一,然而已有算法不能从根本上避免病态解。我们深入分析了联合对角化出现病态解的根本原因,首次把联合对角化问题建模成一个双目标优化模型。在此基础上,我们设计了具有较好条件数的联合对角化算法。该算法给出的对角化器,在最小化对角化误差的同时,还拥有尽可能小的条件数。从而彻底避免了平凡解、不平衡解和退化解。实际上,算法能够给出所给模型Pareto意义下的最优解。此外,该算法对需要对角化的矩阵束几乎没有任何要求。我们也简单讨论了算法的收敛性以及可辨识性。最后,我们还发展了可应用于在线盲分离的联合对角化算法,从而改变了联合对角化算法只适合盲分离批处理算法的现状。实际上,该算法也可以作为一个独立的联合对角化算法,尽管该算法对矩阵束有一些轻微的约束。
3.为时间预测度/方差比方法建立了完整的可分性理论。我们的结果表明源信号在方差比意义下可分当且仅当源信号具有不同的时序结构(即各阶时滞自相关系数),于是,方差比方法的有效性和应用范围得到澄清。该理论还揭示了方差比方法与慢特征提取、二阶统计量方法的内在联系。算法方面,我们将联合对角化技术引入到方差比方法,并且给出了不依赖于源信号的评价指标。最后,我们用数值仿真验证了理论结果的正确性。经过这些努力,方差比方法的可用性和可靠性得到显著增强。
4.提出了在未知源信号数目的情况下估计混叠矩阵的算法。通过一非线性投影,该算法的目标函数的最大值刚好对应混叠矩阵的一列。于是,通过估计目标函数的最大值以及随后的列屏蔽操作,混叠矩阵被逐列估计。因此该算法无须知道源信号的数目(即混叠矩阵的列数)。由于列屏蔽操作会导致大量无用的局部极大值,我们引入粒子群算法优化目标函数。仿真比较结果表明了算法的有效性,特别是在源信号数目未知和源信号不是特别稀疏的情况。
5.提出了基于最小体积约束的非负矩阵分解算法。非负矩阵分解是分离相关源信号的最有力工具之一。我们提出的算法在弱稀疏条件下也能得到理想的结果。我们从理论上分析了非负矩阵分解与稀疏性、最小体积之间的内在联系,解释了模型的合理性和正确性。并提出了两个算法优化目标函数。第一个算法在处理较小规模数据集时非常有效,而第二个算法则更适合处理大数据集问题。仿真表明,我们的算法不仅可以分离一些高度相关的源信号,而且可以显著提高非负矩阵分解学习部分的能力,从而具有非常广泛的用途。
总的说来,本文主要研究了盲分离的线性瞬时模型的相关理论和算法,特别地,对方差比分离方法和应用广泛的联合对角化问题进行了深入的研究。
Blind source separation (BSS) has been found wide applications in many scientific areas, especially in the areas of speech source separation and recognition, biological signal processing, and wireless communication, etc. In fact, due to its high value in theory and promising applications in practice, BSS has been one of the hottest topics in signal processing. Currently, many substantial contributions to BSS have been made. However, some important theoretical and practical issues are still unsolved and they will be focused in this dissertation. The main contributions of this dissertation include:
1. an improvement to the famous minimum-range (MR) geometric BSS method (GBSS). GBSS provides visible separation procedure and geometric interpretation of separation. The MR method is proved reliable theoretically and has no restrictions on the number of observations. However, the optimization algorithm of the method has disadvantages in reliability and efficiency. By virtue of some favorable properties of a convex hull, an improved MR algorithm (iMR) is proposed in this thesis. It can be seen that iMR is more reliable and efficient. Moreover, the maximum-range criterion is also derived to deal with sparse signals, which leads to an extended range of applications of the MR method.
2. a first and in-depth study on the condition number of diagonalizers. Joint diagonalization is one of the most powerful tools to solve BSS problems. However, existing algorithms often can not avoid ill-conditioned solutions strictly. To overcome this disadvantage, the approximate joint diagonalization problem is reviewed as a multi-objective optimization problem for the first time. Then, a new algorithm for nonorthogonal joint diagonalization is developed. The new algorithm yields diagonalizers which have as small condition numbers as possible while minimizing the diagonalization error. Meanwhile, degenerate solutions, trivial solutions and unbalanced solutions are totally avoided strictly. Besides, the new algorithm imposes few restrictions on the target set of matrices to be diagonalized, which makes it widely applicable. Primary results on convergence are presented and we also show that, for exactly jointly diagonalizable sets, no local minima exist and the solutions are essentially unique under mild conditions. The practical use of our algorithm is shown in blind source separation problems, especially when ill-conditioned mixing matrices are involved. Finally, the guided joint diagonalization algorithm (GDiag) is proposed, which is the first applicable joint diagonalization algorithm in online BSS scenario. Indeed, GDiag itself is also a regular joint diagonalization algorithm, although it has some mild restrictions on the set of matrices to be diagonalized.
3. an in-depth separability analysis for the covariance rate method. Our results show that the sources are separable via the covariance rate method if and only if they have different temporal structures (i.e., autocorrelations). Consequently, the applicability and limitations of the covariance rate method are clarified. Also, the relation between the covariance rate method and slow feature analysis, and second-order statistics methods is revealed. In addition, instead of using generalized eigendecomposition, joint approximate diagonalization algorithms are introduced to improve the robustness of the method. A new criterion is presented to evaluate the separation results. Numerical simulations are performed to demonstrate the validity of the theoretical results. With these efforts, the reliability and validity of the covariance rate method is considerably enhanced.
4. a new mixing matrix estimation technique with unknown source number. Although many approaches have been proposed to estimate the mixing matrix, however, they often need to know either the exact value or the upper bound of the source number (i.e. the number of columns of the mixing matrix) as a priori. In this paper, a new method, called NPCM (nonlinear projection and column masking), is proposed to estimate the mixing matrix. In NPCM, the objective function is based on a nonlinear projection such that its maxima just correspond to the columns of the mixing matrix. Then, the columns of the mixing matrix are estimated and deflated sequentially by locating each maximum followed by a masking procedure. As a result, NPCM does not need any prior information on the source number. Because the masking procedure may cause many small and useless local maxima, particle swarm optimization (PSO) is introduced to optimize the objective function. Feasibility and efficiency of PSO are discussed. Comparative experimental results show that NPCM is reasonably competent in the estimation of the mixing matrix, especially in the case that the number of sources is unknown and the sources are less sparse.
5. a new nonnegative matrix factorization (NMF) method to separate highly dependent sources based on minimum-volume constraint (MVC_NMF). NMF is one of the most promising tools to separate statistically dependent sources. The major advantage of our methods is that they can give desirable results even if in very weak sparseness situations. Close relation between NMF and sparseness NMF, MVC_NMF is theoretically analyzed and the reasonability of the new model is justified. Two algorithms are proposed to optimize the objective function. The first one is quite efficient for the relatively small scale problems and the second one is more suitable for larger ones. Simulations show that the new algorithms can not only separate some highly statistically dependent sources but also improve the ability of learning parts of NMF significantly, therefore can be found wide applications in related areas.
To summary, this thesis is focused on the theoretical analysis and algorithms development for the linear instantaneous mixing of BSS. Particularly, we perform in-depth study on the covariance rate method and widely applicable joint diagonalization techniques.
1.改进和推广了著名的盲分离几何算法——最小值域方法。盲分离几何算法能够为盲分离提供可视化的解释和分离过程。其中,最小值域方法有比较严格的理论基础,且对观测信号的个数没有限制。然而,其分离算法的可靠性和效率都存在不足。通过利用凸包的一些优秀性质,本文提出的改进算法效率更高,可靠性更好。另外,本文把最小值域算法推广到最大值域方法,从而扩展了这类几何算法的应用范围。
2.首次对对角化器的条件数展开深入研究。联合对角化是解决盲分离的最重要工具之一,然而已有算法不能从根本上避免病态解。我们深入分析了联合对角化出现病态解的根本原因,首次把联合对角化问题建模成一个双目标优化模型。在此基础上,我们设计了具有较好条件数的联合对角化算法。该算法给出的对角化器,在最小化对角化误差的同时,还拥有尽可能小的条件数。从而彻底避免了平凡解、不平衡解和退化解。实际上,算法能够给出所给模型Pareto意义下的最优解。此外,该算法对需要对角化的矩阵束几乎没有任何要求。我们也简单讨论了算法的收敛性以及可辨识性。最后,我们还发展了可应用于在线盲分离的联合对角化算法,从而改变了联合对角化算法只适合盲分离批处理算法的现状。实际上,该算法也可以作为一个独立的联合对角化算法,尽管该算法对矩阵束有一些轻微的约束。
3.为时间预测度/方差比方法建立了完整的可分性理论。我们的结果表明源信号在方差比意义下可分当且仅当源信号具有不同的时序结构(即各阶时滞自相关系数),于是,方差比方法的有效性和应用范围得到澄清。该理论还揭示了方差比方法与慢特征提取、二阶统计量方法的内在联系。算法方面,我们将联合对角化技术引入到方差比方法,并且给出了不依赖于源信号的评价指标。最后,我们用数值仿真验证了理论结果的正确性。经过这些努力,方差比方法的可用性和可靠性得到显著增强。
4.提出了在未知源信号数目的情况下估计混叠矩阵的算法。通过一非线性投影,该算法的目标函数的最大值刚好对应混叠矩阵的一列。于是,通过估计目标函数的最大值以及随后的列屏蔽操作,混叠矩阵被逐列估计。因此该算法无须知道源信号的数目(即混叠矩阵的列数)。由于列屏蔽操作会导致大量无用的局部极大值,我们引入粒子群算法优化目标函数。仿真比较结果表明了算法的有效性,特别是在源信号数目未知和源信号不是特别稀疏的情况。
5.提出了基于最小体积约束的非负矩阵分解算法。非负矩阵分解是分离相关源信号的最有力工具之一。我们提出的算法在弱稀疏条件下也能得到理想的结果。我们从理论上分析了非负矩阵分解与稀疏性、最小体积之间的内在联系,解释了模型的合理性和正确性。并提出了两个算法优化目标函数。第一个算法在处理较小规模数据集时非常有效,而第二个算法则更适合处理大数据集问题。仿真表明,我们的算法不仅可以分离一些高度相关的源信号,而且可以显著提高非负矩阵分解学习部分的能力,从而具有非常广泛的用途。
总的说来,本文主要研究了盲分离的线性瞬时模型的相关理论和算法,特别地,对方差比分离方法和应用广泛的联合对角化问题进行了深入的研究。
Blind source separation (BSS) has been found wide applications in many scientific areas, especially in the areas of speech source separation and recognition, biological signal processing, and wireless communication, etc. In fact, due to its high value in theory and promising applications in practice, BSS has been one of the hottest topics in signal processing. Currently, many substantial contributions to BSS have been made. However, some important theoretical and practical issues are still unsolved and they will be focused in this dissertation. The main contributions of this dissertation include:
1. an improvement to the famous minimum-range (MR) geometric BSS method (GBSS). GBSS provides visible separation procedure and geometric interpretation of separation. The MR method is proved reliable theoretically and has no restrictions on the number of observations. However, the optimization algorithm of the method has disadvantages in reliability and efficiency. By virtue of some favorable properties of a convex hull, an improved MR algorithm (iMR) is proposed in this thesis. It can be seen that iMR is more reliable and efficient. Moreover, the maximum-range criterion is also derived to deal with sparse signals, which leads to an extended range of applications of the MR method.
2. a first and in-depth study on the condition number of diagonalizers. Joint diagonalization is one of the most powerful tools to solve BSS problems. However, existing algorithms often can not avoid ill-conditioned solutions strictly. To overcome this disadvantage, the approximate joint diagonalization problem is reviewed as a multi-objective optimization problem for the first time. Then, a new algorithm for nonorthogonal joint diagonalization is developed. The new algorithm yields diagonalizers which have as small condition numbers as possible while minimizing the diagonalization error. Meanwhile, degenerate solutions, trivial solutions and unbalanced solutions are totally avoided strictly. Besides, the new algorithm imposes few restrictions on the target set of matrices to be diagonalized, which makes it widely applicable. Primary results on convergence are presented and we also show that, for exactly jointly diagonalizable sets, no local minima exist and the solutions are essentially unique under mild conditions. The practical use of our algorithm is shown in blind source separation problems, especially when ill-conditioned mixing matrices are involved. Finally, the guided joint diagonalization algorithm (GDiag) is proposed, which is the first applicable joint diagonalization algorithm in online BSS scenario. Indeed, GDiag itself is also a regular joint diagonalization algorithm, although it has some mild restrictions on the set of matrices to be diagonalized.
3. an in-depth separability analysis for the covariance rate method. Our results show that the sources are separable via the covariance rate method if and only if they have different temporal structures (i.e., autocorrelations). Consequently, the applicability and limitations of the covariance rate method are clarified. Also, the relation between the covariance rate method and slow feature analysis, and second-order statistics methods is revealed. In addition, instead of using generalized eigendecomposition, joint approximate diagonalization algorithms are introduced to improve the robustness of the method. A new criterion is presented to evaluate the separation results. Numerical simulations are performed to demonstrate the validity of the theoretical results. With these efforts, the reliability and validity of the covariance rate method is considerably enhanced.
4. a new mixing matrix estimation technique with unknown source number. Although many approaches have been proposed to estimate the mixing matrix, however, they often need to know either the exact value or the upper bound of the source number (i.e. the number of columns of the mixing matrix) as a priori. In this paper, a new method, called NPCM (nonlinear projection and column masking), is proposed to estimate the mixing matrix. In NPCM, the objective function is based on a nonlinear projection such that its maxima just correspond to the columns of the mixing matrix. Then, the columns of the mixing matrix are estimated and deflated sequentially by locating each maximum followed by a masking procedure. As a result, NPCM does not need any prior information on the source number. Because the masking procedure may cause many small and useless local maxima, particle swarm optimization (PSO) is introduced to optimize the objective function. Feasibility and efficiency of PSO are discussed. Comparative experimental results show that NPCM is reasonably competent in the estimation of the mixing matrix, especially in the case that the number of sources is unknown and the sources are less sparse.
5. a new nonnegative matrix factorization (NMF) method to separate highly dependent sources based on minimum-volume constraint (MVC_NMF). NMF is one of the most promising tools to separate statistically dependent sources. The major advantage of our methods is that they can give desirable results even if in very weak sparseness situations. Close relation between NMF and sparseness NMF, MVC_NMF is theoretically analyzed and the reasonability of the new model is justified. Two algorithms are proposed to optimize the objective function. The first one is quite efficient for the relatively small scale problems and the second one is more suitable for larger ones. Simulations show that the new algorithms can not only separate some highly statistically dependent sources but also improve the ability of learning parts of NMF significantly, therefore can be found wide applications in related areas.
To summary, this thesis is focused on the theoretical analysis and algorithms development for the linear instantaneous mixing of BSS. Particularly, we perform in-depth study on the covariance rate method and widely applicable joint diagonalization techniques.
引文
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