论文标题
通过广义Schur分解的规范多核分解
Canonical Polyadic Decomposition via the generalized Schur decomposition
论文作者
论文摘要
规范多核分解（CPD）是一种基本张量分解，将张量表示为秩一个张量的总和。与矩阵案例形成鲜明对比的是，使用光假设，低等级张量的CPD（本质上）是唯一的。 CPD的基本唯一性使该分解成为许多应用程序中强大的工具，因为它允许从感兴趣的信号中提取组件信息。 CPD代数计算的一种流行算法是广义特征值分解（GEVD），该分解（GEVD）选择了张量的基质子笔，然后计算铅笔的广义特征向量。在本文中，我们提出了GEVD的简化，以提高算法的准确性。令人惊讶的是，GEVD中的广义特征向量计算实际上是不必要的，可以用QZ分解代替，该分解将一对矩阵作为单一和上三角矩阵的产物。计算QZ分解是计算通用特征向量时的标准第一步，因此我们的算法可以看作是GEVD的直接简化。
The canonical polyadic decomposition (CPD) is a fundamental tensor decomposition which expresses a tensor as a sum of rank one tensors. In stark contrast to the matrix case, with light assumptions, the CPD of a low rank tensor is (essentially) unique. The essential uniqueness of CPD makes this decomposition a powerful tool in many applications as it allows for extraction of component information from a signal of interest. One popular algorithm for algebraic computation of a CPD is the generalized eigenvalue decomposition (GEVD) which selects a matrix subpencil of a tensor, then computes the generalized eigenvectors of the pencil. In this article, we present a simplification of GEVD which improves the accuracy of the algorithm. Surprisingly, the generalized eigenvector computation in GEVD is in fact unnecessary and can be replaced by a QZ decomposition which factors a pair of matrices as a product of unitary and upper triangular matrices. Computing a QZ decomposition is a standard first step when computing generalized eigenvectors, so our algorithm can been seen as a direct simplification of GEVD.
