Global analysis of continuous analogues of the Levenberg-Marquardt and Newton-Raphson methods for solving nonlinear equations

Summary Global analyses are given to continuous analogues of the Levenberg-Marquardt methoddx/dt=−(J ^t(x)J(x)+δI)⁻¹J^t(x)g(x), and the Newton-Raphson-Ben-Israel methoddx/dt=−J ⁺(x)g(x), for solving an over- and under-determined systemg(x)=0 of nonlinear equations. The characteristics of both methods are compared. Erros in some literature which dealt with related continous analogue methods are pointed out. The Institute of Statistical Mathematics     

基于欠定方程的频谱峰值搜索算法及其在连续毫米波雷达系统中的应用

现有谱峰搜索算法在应用于连续毫米波雷达后端中频信号处理时,或计算量较大,或精度不高.本文针对该问题,提出一种可有效提取信号峰值频率信息的快速算法,进而实现连续毫米波雷达的快速准确测速测距.该算法借助压缩感知原理,通过构建欠定方程,并计算少量内积,实现高效捕捉信号频率值.数值结果表明,此算法具有运算量小、计算精度高等特点,可为当今智能交通领域中的高精度测速测距提供一种快速解决方案.     

基于字典尺度自适应学习的欠定盲语音重构算法

针对欠定盲语音分离传统字典学习算法不能优化字典尺寸的问题,提出了一种尺度自适应同步码字优化(Scale Adaptive Simultaneous Codeword Optimization, SASimCO)算法。设计了一种迭代调整字典尺寸的自适应字典学习策略,将训练的字典用于语音盲分离中,以提高语音源信号的恢复性能。所提算法依据设计的候选矩阵,计算候选矩阵中的原子重要性,按照原子重要性准则对字典进行添加与删除原子操作,最后迭代训练得到一个稀疏表示误差最优的字典,用于语音源信号的恢复。使用SiSEC(Signal Separation Evaluation Campaign)数据集对所提算法进行的仿真实验表明,相较于传统字典学习算法,所提算法提高了1～3 dB语音源分离性能,证明了该算法的优势。     

Ill-Conditionedness Need not be Componentwise Near to Ill-Posedness for Least Squares Problems

The condition number of a problem measures the sensitivity of the answer to small changes in the input, where small refers to some distance measure. A problem is called ill-conditioned if the condition number is large, and it is called ill-posed if the condition number is infinity. It is known that for many problems the (normwise) distance to the nearest ill-posed problem is proportional to the reciprocal of the condition number. Recently it has been shown that for linear systems and matrix inversion this is also true for componentwise distances. In this note we show that this is no longer true for least squares problems and other problems involving rectangular matrices. Problems are identified which are arbitrarily ill-conditioned (in a componentwise sense) whereas any componentwise relative perturbation less than 1 cannot produce an ill-posed problem. Bounds are given using additional information on the matrix.     

Fault Feature Extraction for Reciprocating Compressors Based on Underdetermined Blind Source Separation

In practical engineering applications, the vibration signals collected by sensors often contain outliers, resulting in the separation accuracy of source signals from the observed signals being seriously affected. The mixing matrix estimation is crucial to the underdetermined blind source separation (UBSS), determining the accuracy level of the source signals recovery. Therefore, a two-stage clustering method is proposed by combining hierarchical clustering and K-means to improve the reliability of the estimated mixing matrix in this paper. The proposed method is used to solve the two major problems in the K-means algorithm: the random selection of initial cluster centers and the sensitivity of the algorithm to outliers. Firstly, the observed signals are clustered by hierarchical clustering to get the cluster centers. Secondly, the cosine distance is used to eliminate the outliers deviating from cluster centers. Then, the initial cluster centers are obtained by calculating the mean value of each remaining cluster. Finally, the mixing matrix is estimated with the improved K-means, and the sources are recovered using the least square method. Simulation and the reciprocating compressor fault experiments demonstrate the effectiveness of the proposed method.     

稀疏分量分析在欠定语音信号盲分离中的应用

研究了基于两步法的欠定语音信号盲分离。针对混合信号散点图在原点中心混叠程度过高的缺点,提出了弭灭圆K均值聚类算法,提高了混叠矩阵的估计精度。结合时频分析算法实现了欠定瞬时线性混叠语音信号的盲分离,取得了较好的分离效果。     

Bounds on smooth solutions of underdetermined linear models for gas exchange

In recent literature, grossly underdetermined linear models have been developed for certain experimental techniques in respiratory physiology. Because physiologically sensible solutions of the models are reasonably assumed to be smooth, it is possible to recover such solutions using smoothed least-squares techniques. Bounds on the variability of smooth solutions satisfying the model can be generated using linear programming.The second author was supported in this work by New Zealand Medical Research Council Grant No. 81/65 to Dr. E. A. Harris, Greenlane Hospital, Auckland, New Zealand. The authors are indebted to Dr. Harris for his support and guidance in the physiological aspects of the work.     

基于比率矩阵聚类的欠定盲分离算法

利用稀疏分量的直线聚类性,提出了欠定盲源分离中估计混合矩阵的一种方法。该方法通过构造比率矩阵对观测信号进行分选,剔除了源信号频谱重叠的部分,然后利用鲁棒竞争的聚类学习算法获得对混合矩阵的精确估计,解决了源信号在频域不充分稀疏的条件下准确估计混合矩阵的问题。在估计出混合矩阵的基础上,利用最短路径法分离出源信号。由仿真结果可以看出,与传统的K均值估计混合矩阵的方法相比,方法具有更好的鲁棒性。     

基于ICA的盲信号分离正定性检验方法

 盲信号分离中,判断观测信号个数与实际信源个数的关系对于信号分离算法的选择和算法效果的评估非常重要,但目前还缺乏有效的方法对正定和欠定情况进行区分性检验.针对这一问题,本文提出两种检验方法.第一种方法通过分析整个数据序列ICA分解输出分量之间的独立性来实现.理论分析表明,欠定条件下ICA分解输出分量之间必然不具有独立性,而正定条件下,只要源信号满足独立的前提假设,ICA分解输出分量之间可以相互独立.第二种方法中,我们对数据序列不同位置添加等长时间窗,根据ICA分解基向量的稳定性来检验正定性.理论分析表明,正定条件下不同窗口数据ICA分解的基向量都收敛到混合阵的基向量上,而欠定条件下,分解得到的基向量随不同时窗内源信号分布的变化而改变.本文通过仿真实验,证明了这两种方法的可行性.     

基于孤立点检测的欠定混合矩阵盲辨识

本文将孤立点检测的思想引入到欠定混合矩阵的盲辨识问题,提出了一种基于孤立点检测的混合矩阵盲辨识方法.首先计算混合信号的空间时频分布并检测出单源时频点,然后检测出单源时频点中的孤立点并将其从中去除,再通过聚类的方法估计混合矩阵.该方法降低了对信号稀疏性的要求,通过去除数据中的孤立点,提高了矩阵的估计精度,同时也有助于对源信号数目的估计.仿真实验表明,与已有算法相比,本文方法进一步提高了混合矩阵的估计精度,并且有更强的鲁棒性.