线性混合盲信源分离的算法研究
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摘要
在源信号和传输信道均未知的情况下,仅利用接收天线的观测信号提取或者恢复相互统计独立/不相关的源信号的过程,称为盲信源分离。盲信源分离的信号模型具有一般性,因此它在生物医学信号处理、音频和语音信号处理、多用户通信以及数据分析等领域,具有非常广阔的应用前景,引起了信号处理和神经网络等领域的专家和学者的广泛关注。本文围绕解决线性混合盲信源分离问题的代数方法和对比函数优化方法,着重研究了联合(块)对角化算法、张量分解算法以及自然梯度算法,并对相关信源盲分离问题进行了初步探索。
论文的主要创新性成果总结如下:
1.非正交联合对角化避免了预白化所导致的性能下降,但其解不一定唯一存在,论文提出并证明了非正交联合对角化盲分离算法的可辨识性条件。论文给出了非正交联合对角化解唯一存在的定义,并给出了解唯一存在的条件;从该条件出发分别推导得到了基于观测信号的二阶时延相关函数和高阶累积量函数的非正交联合对角化算法实现盲分离时对源信号统计特性的要求。论文的结论明确了非正交联合对角化盲分离算法的适用范围,从而进一步完善了该算法。
2.通过对联合对角化的雅克比方法加以改造,论文提出了一种基于GIVENS旋转的正交联合块对角化算法。将GIVENS矩阵中参数的选择转化为一元四次三角函数多项式的优化问题,并调整旋转的循环顺序,即可实现正交联合块对角化。将提出的正交联合块对角化算法和预白化结合起来可以有效解决多维盲信源分离和线性卷积混合的盲信源分离问题。
3.通过引入张量的塔克分解,提出了一种张量标准分解的快速算法,并将其应用于盲源分离欠定混合矩阵的估计问题中。张量标准分解的因子矩阵和欠定混合矩阵的估计一样,也存在幅值和排列顺序的不确定性,从而可以把欠定混合矩阵的估计转换为张量的标准分解问题。为了克服原有的标准分解算法运算复杂度高、所需时间长的缺点,论文引入张量的塔克分解将待分解张量压缩为维数更低的核张量,塔克分解因子可通过原张量mode-3矩阵的左奇异向量求得,然后运用交替最小二乘对该核张量进行标准分解,即可得到混合矩阵的估计。论文算法有效地降低了估计欠定混合矩阵的张量分解算法的运算量。
4.激励函数估计的准确性是影响自然梯度算法收敛速度和稳态性能的一个重要因素,论文提出了一种采用函数逼近直接估计激励函数的方法。该方法用一组正交多项式的线性组合来逼近激励函数,逼近的程度用均方误差来衡量。线性组合的系数向量可运用分值函数的性质,通过最小化均方误差自适应学习得到。采用论文方法估计激励函数可使自然梯度算法获得更快的收敛速度。
5.相关信源混合的盲分离问题是当前研究的难点,针对具备时域稀疏性的相关信源线性混合问题,论文提出了一种稀疏相关信源盲分离算法。对于在某些时刻只有一个源信号处于活跃状态的相关信源盲分离问题,一个源信号单独存在时刻的观测信号向量正是对混合矩阵某一列的估计,可以利用这一性质实现对混合矩阵的估计;一个源信号单独存在的时刻可以通过对两传感器观测信号比值的方差进行比较而确定。该算法为解决源信号具备稀疏性的相关信源盲分离问题提供了一个很好的思路。
Blind source separation(BSS) consists of recovering mutually independent/ uncorrelated but otherwise unobserved source signals from their mixtures without any prior knowledge of the channel. BSS has attracted growing attention in statistical signal processing and unsupervised neural learning society, since it is a fundamental problem encountered in various fields, such as biomedical signal processing, audio and acoustics, multi-user communications, data analysis, and so on. In this dissertation, we focus on the method of algebra and optimizing contrast function to linear mixture BSS, and we investigate the BSS algorithms of joint diagonalization(JD), joint block diagonalization(JBD), tensor decomposition and natural gradient. In addition, some preliminary exploration on BSS of correlated sources are carried out.
The primary contributions included in this dissertation are summarized blow:
1. Non-orthogonal joint diagonalization could avoid the performance degradation caused by pre-whitening, but the solution is not necessarily only. We analyzed the identifiability for nonorthogonal joint diagonalization(NJD). We proposed the uniqueness condition of the solution to NJD, and pointed out that uniqueness condition for NJD is that the vectors consisting of diagonal elements in the same position of diagonal matrix are pairwise linearly independent. From this proposition,the necessary and sufficient condition for BSS is deduced. For second-order statistics based BSS, the condition is that the source signals have not the identical autocorrelation shape. For higher-order cumulant, there is not Gaussian signal in sources. The above conclusion provides a mathematical foundation for the BSS methods based on the NJD.
2. We proposed an algorithm of orthogonal joint block diagonalization based on GIVENS rotation. In this method, we improved Jacobi algorithm which perfect solve the problem of orthogonal joint diagonalization. We transformed the selection of GIVENS matrix’s parameter into the optimization of univariate quadruplicate trigonometric function and adjusted the cycle of rotation. So the new algorithm could solve the problem of orthogonal JBD. The new algorithm combination with pre-whitening procedure can solve Multidimensional blind source separation(MBSS) and convolutive BSS.
3. In this paper, a fast algorithm of tensor canonical decomposition is proposed through tucker decomposition.Same as the estimation of underdetermined mixture matrix, there are permutation and scaling indeterminacies in factor matrix of canonical decomposition. So the estimation of underdetermined mixture matrix can be performed through tensor canonical decomposition. In order to overcome the flaw of high computational complexity and long running time of existing canconical decomposition algorithm,compress the tensor into lower order core one using tucker decomposition.The factor of tucker decomposition can be obtained by left singular value of the original tensor’s mode-3 matrix. The mixture matrix can be estimated by the alternating least squares based canonical decomposition of the core tensor. The proposed algorithm has much lower computational complexity with no performance loss than existing algorithm.
4. The accuracy of activation function is an important factor that influences the convergent speed and stability of the natural gradient algorithm. A new estimating activation function approach based on the method of function approximation is proposed. In this approach, activation function was approximated by the linear combination of a set of orthogonal polynomials. The accuracy of approximating is measured by mean square error(MSE). Using the property of score function ,coefficients of the linear combination can be obtained by adaptive minimizing MSE. The new BSS algorithm is developed by substituting the estimated activation function into natural gradient iterative formula. Compared with the traditional one, convergent speed of the new algorithm is highly improved.
5. Dependent sources BSS is a difficult problem in this research field. We proposed an algorithm which could resolve the problem of dependent sources BSS of time domain sparseness. For BSS of sparse souces, there are some instants at which only one source existing.The vectors of observation at these instants are the estimation of the corresponding columns at mixing matrix. Using this property, mixing matrix can be estimated correctly. These instants can be obtained by comparing variances of the ratio between two sensors. By this special way, the sparse dependent BSS problem can be resolved.
论文的主要创新性成果总结如下:
1.非正交联合对角化避免了预白化所导致的性能下降,但其解不一定唯一存在,论文提出并证明了非正交联合对角化盲分离算法的可辨识性条件。论文给出了非正交联合对角化解唯一存在的定义,并给出了解唯一存在的条件;从该条件出发分别推导得到了基于观测信号的二阶时延相关函数和高阶累积量函数的非正交联合对角化算法实现盲分离时对源信号统计特性的要求。论文的结论明确了非正交联合对角化盲分离算法的适用范围,从而进一步完善了该算法。
2.通过对联合对角化的雅克比方法加以改造,论文提出了一种基于GIVENS旋转的正交联合块对角化算法。将GIVENS矩阵中参数的选择转化为一元四次三角函数多项式的优化问题,并调整旋转的循环顺序,即可实现正交联合块对角化。将提出的正交联合块对角化算法和预白化结合起来可以有效解决多维盲信源分离和线性卷积混合的盲信源分离问题。
3.通过引入张量的塔克分解,提出了一种张量标准分解的快速算法,并将其应用于盲源分离欠定混合矩阵的估计问题中。张量标准分解的因子矩阵和欠定混合矩阵的估计一样,也存在幅值和排列顺序的不确定性,从而可以把欠定混合矩阵的估计转换为张量的标准分解问题。为了克服原有的标准分解算法运算复杂度高、所需时间长的缺点,论文引入张量的塔克分解将待分解张量压缩为维数更低的核张量,塔克分解因子可通过原张量mode-3矩阵的左奇异向量求得,然后运用交替最小二乘对该核张量进行标准分解,即可得到混合矩阵的估计。论文算法有效地降低了估计欠定混合矩阵的张量分解算法的运算量。
4.激励函数估计的准确性是影响自然梯度算法收敛速度和稳态性能的一个重要因素,论文提出了一种采用函数逼近直接估计激励函数的方法。该方法用一组正交多项式的线性组合来逼近激励函数,逼近的程度用均方误差来衡量。线性组合的系数向量可运用分值函数的性质,通过最小化均方误差自适应学习得到。采用论文方法估计激励函数可使自然梯度算法获得更快的收敛速度。
5.相关信源混合的盲分离问题是当前研究的难点,针对具备时域稀疏性的相关信源线性混合问题,论文提出了一种稀疏相关信源盲分离算法。对于在某些时刻只有一个源信号处于活跃状态的相关信源盲分离问题,一个源信号单独存在时刻的观测信号向量正是对混合矩阵某一列的估计,可以利用这一性质实现对混合矩阵的估计;一个源信号单独存在的时刻可以通过对两传感器观测信号比值的方差进行比较而确定。该算法为解决源信号具备稀疏性的相关信源盲分离问题提供了一个很好的思路。
Blind source separation(BSS) consists of recovering mutually independent/ uncorrelated but otherwise unobserved source signals from their mixtures without any prior knowledge of the channel. BSS has attracted growing attention in statistical signal processing and unsupervised neural learning society, since it is a fundamental problem encountered in various fields, such as biomedical signal processing, audio and acoustics, multi-user communications, data analysis, and so on. In this dissertation, we focus on the method of algebra and optimizing contrast function to linear mixture BSS, and we investigate the BSS algorithms of joint diagonalization(JD), joint block diagonalization(JBD), tensor decomposition and natural gradient. In addition, some preliminary exploration on BSS of correlated sources are carried out.
The primary contributions included in this dissertation are summarized blow:
1. Non-orthogonal joint diagonalization could avoid the performance degradation caused by pre-whitening, but the solution is not necessarily only. We analyzed the identifiability for nonorthogonal joint diagonalization(NJD). We proposed the uniqueness condition of the solution to NJD, and pointed out that uniqueness condition for NJD is that the vectors consisting of diagonal elements in the same position of diagonal matrix are pairwise linearly independent. From this proposition,the necessary and sufficient condition for BSS is deduced. For second-order statistics based BSS, the condition is that the source signals have not the identical autocorrelation shape. For higher-order cumulant, there is not Gaussian signal in sources. The above conclusion provides a mathematical foundation for the BSS methods based on the NJD.
2. We proposed an algorithm of orthogonal joint block diagonalization based on GIVENS rotation. In this method, we improved Jacobi algorithm which perfect solve the problem of orthogonal joint diagonalization. We transformed the selection of GIVENS matrix’s parameter into the optimization of univariate quadruplicate trigonometric function and adjusted the cycle of rotation. So the new algorithm could solve the problem of orthogonal JBD. The new algorithm combination with pre-whitening procedure can solve Multidimensional blind source separation(MBSS) and convolutive BSS.
3. In this paper, a fast algorithm of tensor canonical decomposition is proposed through tucker decomposition.Same as the estimation of underdetermined mixture matrix, there are permutation and scaling indeterminacies in factor matrix of canonical decomposition. So the estimation of underdetermined mixture matrix can be performed through tensor canonical decomposition. In order to overcome the flaw of high computational complexity and long running time of existing canconical decomposition algorithm,compress the tensor into lower order core one using tucker decomposition.The factor of tucker decomposition can be obtained by left singular value of the original tensor’s mode-3 matrix. The mixture matrix can be estimated by the alternating least squares based canonical decomposition of the core tensor. The proposed algorithm has much lower computational complexity with no performance loss than existing algorithm.
4. The accuracy of activation function is an important factor that influences the convergent speed and stability of the natural gradient algorithm. A new estimating activation function approach based on the method of function approximation is proposed. In this approach, activation function was approximated by the linear combination of a set of orthogonal polynomials. The accuracy of approximating is measured by mean square error(MSE). Using the property of score function ,coefficients of the linear combination can be obtained by adaptive minimizing MSE. The new BSS algorithm is developed by substituting the estimated activation function into natural gradient iterative formula. Compared with the traditional one, convergent speed of the new algorithm is highly improved.
5. Dependent sources BSS is a difficult problem in this research field. We proposed an algorithm which could resolve the problem of dependent sources BSS of time domain sparseness. For BSS of sparse souces, there are some instants at which only one source existing.The vectors of observation at these instants are the estimation of the corresponding columns at mixing matrix. Using this property, mixing matrix can be estimated correctly. These instants can be obtained by comparing variances of the ratio between two sensors. By this special way, the sparse dependent BSS problem can be resolved.
引文
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