稀疏盲分离的理论与算法研究

发布时间：2018-05-09 08:57

本文选题：盲分离 + 稀疏表示　；参考：《广东工业大学》2016年博士论文

【摘要】：盲信号分离在众多科学领域有着重要的研究,特别是语音识别、无线通讯、信号抗干扰、图像处理、特征抽取、生物医学、经济学、声呐、遥感图像解译、地震探测信号处理、信号加密等领域,有着极其广泛的应用,因此受到越来越多学者的关注,涌现了大量优秀的盲分离理论与算法。然而,随着盲分离领域的快速发展,同时也暴露出一些需要进一步解决的关键性理论与实际问题。如怎样进行稀疏盲源信号分离?经典的FOCUSS算法的收敛性和收敛率问题?在源信号不充分稀疏的情况下,如何借助非负的条件进行盲源分离等等。本博士论文将围绕以下几个方面继续探讨稀疏盲源信号分离问题：首先对FOCUSS算法给出了严格的理论推导并证明了其收敛性。基于给定的4个假设,通过借助辅助函数对FOCUSS算法作严格推导,并针对已有的FOCUSS算法推导中存在的理论问题,给出了全新而严格的FOCUSS算法的数学推导,并以此通过稳定性分析进一步证明FOCUS S算法的收敛性。通过大量的实验发现,即使面对大规模的数据集(例如,M=3000和N=5000),FOCUSS算法可以在迭代次数小于50时找到最优解。收敛率是考查算法性能的另一个关键指标。继FOUCSS算法的收敛性证明后,我们继续系统地研究FOCUSS算法在参数p∈(0,2)区间的收敛率。经研究发现：当0p1时,FOUCSS算法超线性收敛,且其收敛阶数为r=2-p：当1≤p2时,FOUCSS算法通常会线性收敛；然而,对于1≤p2条件的某些特殊情形,FOUCSS算法可能出现超线性收敛。同时,我们还给出了以上各种情况出现的条件和相应的仿真结果,并同步证明了以上各种情形下的收敛率。然后,提出了多层FOCUSS算法,大幅度提升了稀疏表示信号重构精度。当p1时,理论上FOCUSS算法可得到更为精确的稀疏表示结果。然而在实际操作中,对于p1情形,FOCUSS算法通常会因陷入局部极小值从而导致效果很不理想。此类问题借助我们提出的多层FOCUSS算法来求解。相比常规FOCUSS算法,随着层数增加,多层FOCUSS算法可逐步提高信号重构精度,最终精度大幅度优于常规FOCUSS算法。此外,还提出了更鲁棒的盲辨识新算法。因为利用FOCUSS算法恢复稀疏源信号依赖于混叠矩阵的精确估计,所以混的矩阵的精确估计也是人们关注的问题,尽管平行因子分析(或者标准多元分解)算法是估计混叠矩阵最为有效的工具之一,但现有的平行因子分析算法容易陷入局部收敛,从而严重影响盲辨识和盲分离的精度。通过整合交替最小二乘(ALS)的方法和分层ALS(HALS)方法的优点,我们提出一类基于并行秩-1的奇异值分解的新算法,新算法有着高度并行能力,并且很大程度上避免了局部最小的发生。还有,在源信号不充分稀疏的情况下,我们还提出多边锥拟合的非负矩阵分解方法(NMF)。研究了可分假设和k-稀疏之间的联系,并验证了可分假定等价于k=1情形下的k-稀疏条件；借助半平面辨识,我们提出了几种新的NMF算法,这些算法只需通过简单的特征值分解(EVD)即可实现。与Gillis和Vava is的方法相比,在新算法中,1-稀疏条件可以放宽到更弱的(m-1)-稀疏；最后通过实验证明发现,提出的算法在计算精确度方面大大优于当前最先进的算法。再有,提出了非线性局部平滑约束的NMF(NMF-NCR)的方法,并用此方法来处理谱分解问题。方法使用了交替迭代优化方案,即,固定一个因子矩阵来优化另一个因子矩阵。在优化每个因子矩阵时,证明了代价函数的梯度是Lipschitz连续的。据此设计了一个近似函数来优化原始代价函数。该算法具有非线性收敛率,比传统具有线性收敛率的方法速度快得多。最后,对全文的工作进行总结。
[Abstract]:Blind signal separation has important research in many fields of science, especially in the fields of speech recognition, wireless communication, signal anti-jamming, image processing, feature extraction, biomedical, economics, sonar, remote sensing image interpretation, seismic signal processing, signal encryption and so on, so it has been paid more and more attention by more and more scholars. There have been a large number of excellent blind separation theories and algorithms. However, with the rapid development of the blind separation field, some key theoretical and practical problems are exposed. For example, how to separate the sparse blind source signal, the convergence and the convergence rate of the classical FOCUSS algorithm and the lack of sparse source signal In this paper, we will continue to discuss the problem of Sparse Blind Source Separation in the following aspects: first, we give a strict theoretical deduction to the FOCUSS algorithm and prove its convergence. Based on the given 4 assumptions, the FOCUSS algorithm is strictly pushed through the aid of the auxiliary function. In view of the existing theoretical problems in the derivation of the existing FOCUSS algorithm, a new and strict mathematical derivation of the FOCUSS algorithm is given. Through the stability analysis, the convergence of the FOCUS S algorithm is further proved. Through a large number of experiments, it is found that even in the face of large numbers of data sets (such as M=3000 and N=5000), the FOCUSS algorithm can be used in a large number of experiments. The optimal solution is found when the number of iterations is less than 50. The convergence rate is another key index for the performance of the algorithm. After the convergence of the FOUCSS algorithm, we continue to systematically study the convergence rate of the FOCUSS algorithm in the parameter P 0,2 (0,2) interval. It is found that when 0p1, the FOUCSS algorithm is superlinear and its convergence order is r=2-p: when 1 < < When P2, the FOUCSS algorithm usually converges linearly; however, the FOUCSS algorithm may have superlinear convergence for some special cases of 1 < P2 condition. At the same time, we give the conditions and the corresponding simulation results of all the above cases, and synchronously prove the convergence rate under all kinds of circumstances. Then, the multi-layer FOCUSS algorithm is proposed. The precision of sparse representation signal reconstruction is greatly improved. When P1, the FOCUSS algorithm can theoretically get more accurate sparse representation results. However, in the actual operation, for the case of P1, the FOCUSS algorithm usually results in the local minima resulting in the effect is not ideal. This question is solved by the multi-layer FOCUSS algorithm proposed by us. Compared with the conventional FOCUSS algorithm, with the increase of the number of layers, the multi-layer FOCUSS algorithm can improve the precision of the signal reconstruction step by step, and the final precision is much better than the conventional FOCUSS algorithm. In addition, a more robust blind identification algorithm is proposed. Because the FOCUSS algorithm is used to restore the sparse source signal to the accurate estimation of the aliasing matrix, the mixture matrix is fine. It is also a problem that people pay attention to. Although parallel factor analysis (or standard multivariate decomposition) algorithm is one of the most effective tools to estimate the aliasing matrix, the existing parallel factor analysis algorithm is easy to fall into local convergence, thus seriously affecting the accuracy of blind identification and blind separation. By integrating the method of alternating least squares (ALS) and The advantages of layered ALS (HALS) method, we propose a new algorithm for singular value decomposition based on parallel rank -1. The new algorithm has high parallel ability and avoids the occurrence of local minimum to a large extent. In the case of insufficient source signal, we also propose a non negative matrix decomposition method (NMF) for multilateral cone fitting (NMF). The relation between the separable hypothesis and the k- sparsity is presented and the k- sparse condition of the separable hypothesis equivalent to the k=1 case is verified. With the help of the semi plane identification, we propose several new NMF algorithms, which only need to be realized by simple eigenvalue decomposition (EVD). In comparison with the Gillis and Vava is methods, the 1- sparsity conditions are in the new algorithm. It can be relaxed to the weaker (m-1) - sparsity. Finally, the experimental results show that the proposed algorithm is much better than the most advanced algorithm in computational accuracy. Then, the method of NMF (NMF-NCR) for nonlinear local smooth constraints is proposed, and the method is used to deal with the spectral decomposition problem. A factor matrix is fixed to optimize another factor matrix. When each factor matrix is optimized, the gradient of the cost function is proved to be Lipschitz continuous. According to this, an approximate function is designed to optimize the original cost function. The algorithm has a nonlinear convergence rate, much faster than the traditional method of wired convergence. The full text of the work is summarized.

【学位授予单位】：广东工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TN911.7

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