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Joint Eigenvalue Decomposition of Non-Defective Matrices Based on the LU Factorization With Application to ICA

机译:基于LU分解的无缺陷矩阵联合特征值分解及其在ICA中的应用

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In this paper we propose a fast and efficient Jacobi-like approach named JET (Joint Eigenvalue decomposition based on Triangular matrices) for the Joint EigenValue Decomposition (JEVD) of a set of real or complex non-defective matrices based on the LU factorization of the matrix of eigenvectors. Contrarily to classical Jacobi-like JEVD methods, the iterative procedure of the JET approach can be reduced to the search for only one of the two triangular matrices involved in the factorization of the matrix of eigenvectors, hence decreasing the numerical complexity. Two variants of the JET technique, namely JET-U and JET-O, which correspond to the optimization of two different cost functions are described in detail and these are extended to the complex case. Numerical simulations show that in many practical cases the JET approach provides more accurate estimation of the matrix of eigenvectors than its competitors and that the lowest numerical complexity is consistently achieved by the JET-U algorithm. In addition, we illustrate in the ICA context the interest of being able to solve efficiently the (non-orthogonal) JEVD problem. More particularly, based on our JET-U algorithm, we propose a more robust version of an existing ICA method, named MICAR-U. The identifiability of the latter is studied and proved under some conditions. Computer results given in the context of brain interfaces show the better ability of MICAR-U to denoise simulated electrocortical data compared to classical ICA techniques for low signal to noise ratio values.
机译:在本文中,我们提出了一种快速有效的基于Jacobi的方法,称为JET(基于三角矩阵的联合特征值分解),用于基于矩阵LU分解的一组实或复杂无缺陷矩阵的联合特征值分解(JEVD)。特征向量矩阵。与经典的Jacobi型JEVD方法相反,JET方法的迭代过程可以简化为仅搜索与特征向量矩阵分解有关的两个三角形矩阵之一,从而降低了数值复杂度。详细描述了JET技术的两个变体,即JET-U和JET-O,它们对应于两个不同成本函数的优化,并将它们扩展到复杂的情况。数值模拟表明,在许多实际情况下,JET方法比其竞争对手提供了更准确的特征向量矩阵估计,并且JET-U算法始终实现最低的数值复杂性。另外,我们在ICA上下文中说明了能够有效解决(非正交)JEVD问题的兴趣。更特别地,基于我们的JET-U算法,我们提出了现有ICA方法的更强大版本,称为MICAR-U。在一定条件下研究并证明了后者的可识别性。与传统的ICA技术相比,对于低信噪比值,在大脑界面环境下给出的计算机结果显示MICAR-U具有更好的去噪模拟电皮质数据的能力。

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