Iterative methods for the canonical decomposition of multi-way arrays: Application to blind underdetermined mixture identification
Iterative methods for the canonical decomposition of multi-way arrays: Application to blind underdetermined mixture identification
复制标题
DOI：
10.1016/j.sigpro.2011.02.003
复制
发表时间：
2011-08
期刊：
影响因子：
--
通讯作者：
A. Karfoul;L. Albera;L. D. Lathauwer
中科院分区：
文献类型：
--
作者：
A. Karfoul;L. Albera;L. D. Lathauwer
交替最小二乘（ALS）算法用于拟合多路阵列的典型分解（CAND），有两个主要缺点。首先，它的缓慢收敛所造成的存在的因素之间的共线性的多路阵列，它分解。其次，它的盲目性的厄米对称性的考虑阵列。增强线搜索（ELS）计划被发现是一个很好的方式来科普的ALS算法的收敛速度慢，连同部分使用的厄米特对称性。然而，据我们所知，所需的方程，以执行后者计划的情况下，只有三阶和五阶阵列。因此，我们的第一个贡献在于推广的ELS程序的情况下，复杂的数组的任何顺序大于三。我们的第二个贡献是另一个改进的ALS计划，能够受益于Hermitianity和半正定的考虑数组。它包括诉诸CAND第一个三阶阵列具有一个酉加载矩阵和第二个几个秩1阵列。然后提出了一种迭代算法之间交替的Procrustes问题的解决和秩一矩阵近似的计算，以实现三阶阵列的CAND。
Two main drawbacks can be stated in the alternating least square (ALS) algorithm used to fit the canonical decomposition (CAND) of multi-way arrays. First its slow convergence caused by the presence of collinearity between factors in the multi-way array it decomposes. Second its blindness to Hermitian symmetries of the considered arrays. Enhanced line search (ELS) scheme was found to be a good way to cope with the slow convergence of the ALS algorithm together with a partial use of the Hermitian symmetry. However, to our knowledge, required equations to perform the latter scheme are only given in the case of third and fifth order arrays. Therefore, our first contribution consists in generalizing the ELS procedure to the case of complex arrays of any order greater than three. Our second contribution is another improvement of the ALS scheme, able to profit from Hermitianity and positive semi-definiteness of the considered arrays. It consists in resorting to the CAND first of a third order array having one unitary loading matrix and second of several rank-1 arrays. An iterative algorithm is then proposed alternating between Procrustes problem solving and the computation of rank-one matrix approximations in order to achieve the CAND of the third order array.