In this paper, we propose a new approach to compute the Joint EigenValue Decomposition (JEVD) of real or complex matrix sets. JEVD aims to find a common basis of eigenvectors to a set of matrices. JEVD problem is encountered in many signal processing applications. In particular, recent and efficient algorithms for the Canonical Polyadic Decomposition (CPD) of multiway arrays resort to a JEVD step. The suggested method is based on multiplicative updates. It is distinguishable by the use of a first-order Taylor Expansion to compute the inverse of the updating matrix. We call this approach Joint eigenvalue Decomposition based on Taylor Expansion (JDTE). This approach is derived in two versions based on simultaneous and sequential optimization schemes respectively. Here, simultaneous optimization means that all entries of the updating matrix are simultaneously optimized at each iteration. To the best of our knowledge, such an optimization scheme had never been proposed to solve the JEVD problem in a multiplicative update procedure. Our numerical simulations show that, in many situations involving complex matrices, the proposed approach improves the eigenvectors estimation while keeping a limited computational cost. Finally, these features are highlighted in a practical context of source separation through the CPD of telecommunication signals.

中文翻译：

---

基于一阶泰勒展开式的联合特征值分解算法


在本文中，我们提出了一种计算实数或复数矩阵集的联合特征值分解（JEVD）的新方法。 JEVD 旨在找到一组矩阵的特征向量的共同基础。许多信号处理应用中都会遇到 JEVD 问题。特别是，多路阵列的正则多元分解 (CPD) 的最新高效算法采用了 JEVD 步骤。建议的方法基于乘法更新。它可以通过使用一阶泰勒展开来计算更新矩阵的逆来区分。我们将这种方法称为基于泰勒展开的联合特征值分解（JDTE）。该方法有两个版本，分别基于同时优化方案和顺序优化方案。这里，同时优化意味着更新矩阵的所有条目在每次迭代时同时优化。据我们所知，从未提出过这样的优化方案来解决乘法更新过程中的 JEVD 问题。我们的数值模拟表明，在许多涉及复杂矩阵的情况下，所提出的方法改进了特征向量估计，同时保持有限的计算成本。最后，这些特征在通过电信信号的 CPD 进行源分离的实际背景下得到强调。