A semi-algebraic framework for approximate CP decompositions via simultaneous matrix diagonalizations (SECSI)
A semi-algebraic framework for approximate CP decompositions via simultaneous matrix diagonalizations (SECSI)
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DOI：
10.1016/j.sigpro.2013.02.016
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发表时间：
2013-09
期刊：
影响因子：
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通讯作者：
F. Roemer;M. Haardt
中科院分区：
文献类型：
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作者：
F. Roemer;M. Haardt
在本文中，我们提出了一个框架来计算近似CANDECOMP / PARAFAC（CP）分解。这种张量分解在广泛的应用中是可行的工具，需要通用的工具来计算这种分解，并具有可调整的复杂性-准确性权衡。为此，我们提出了一种新的SEMI代数框架，允许通过同时矩阵对角化（SECSI）的近似CP分解的计算。与以前的同时矩阵对角化（SMD）为基础的方法相比，我们使用张量结构来构建不仅是一个，但可能的SMD的完整集合。解决所有的SMD，我们得到多个估计的因素矩阵和目前的战略，以选择最佳的估计在随后的步骤。该SECSI框架保留了选择要求解的SMD数量的选项，并采用各种策略从多个估计中选择最终解决方案。最佳匹配方案的基础上的穷举搜索以及启发式选择方案的设计，以灵活地适应特定的应用。四个例子的算法与不同的精度-复杂度权衡点进行比较，以国家的最先进的算法。我们得到更可靠的估计和降低计算复杂性。
In this paper, we propose a framework to compute approximate CANDECOMP / PARAFAC (CP) decompositions. Such tensor decompositions are viable tools in a broad range of applications, creating the need for versatile tools to compute such decompositions with an adjustable complexity-accuracy trade-off. To this end, we propose a novel SEmi-algebraic framework that allows the computation of approximate C P decompositions via SImultaneous Matrix Diagonalizations (SECSI). In contrast to previous Simultaneous Matrix Diagonalization (SMD)-based approaches, we use the tensor structure to construct not only one but the full set of possible SMDs. Solving all SMDs, we obtain multiple estimates of the factor matrices and present strategies to choose the best estimate in a subsequent step. This SECSI framework retains the option to choose the number of SMDs to solve and to adopt various strategies for the selection of the final solution out of the multiple estimates. A best matching scheme based on an exhaustive search as well as heuristic selection schemes are devised to flexibly adapt to specific applications. Four example algorithms with different accuracy-complexity trade-off points are compared to state-of-the-art algorithms. We obtain more reliable estimates and a reduced computational complexity.