SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 832-858, January 2021.
In practical applications, incremental tensors are very common: only a portion of tensor data is available, and new data are arriving in the next time step or continuously over time. To handle this type of tensors time-saving algorithms are required for online computation. In this paper, we consider incremental CP (CANDECOMP/PARAFAC) decomposition, which requires one to update the CP decomposition after new tensor data, together with the exiting tensor, are ready for analysis. There exist several incremental CP decomposition algorithms, but almost all of these algorithms assume that the number of CP decomposition components remains fixed in the incremental process. The main contribution of this paper is the study of how to add components in the incremental CP decomposition. We derive the coordinate representation of the incremental CP decomposition with respect to a special basis and show related properties of tensor rank and the uniqueness of such incremental CP decomposition. Under the framework of this representation, the proposed method can be solved by using an alternating minimization algorithm. Numerical examples are presented to show the good performance of the proposed algorithms in terms of computational time and data fitting compared with existing methods.

中文翻译：

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交替最小化法增量CP张量分解

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 832-858 页，2021 年 1 月。
在实际应用中，增量张量非常常见：只有一部分张量数据可用，新数据在下一个时间步或随着时间的推移不断地到达。为了处理这种类型的张量，在线计算需要节省时间的算法。在本文中，我们考虑增量 CP（CANDECOMP/PARAFAC）分解，它需要在新张量数据和现有张量准备好分析后更新 CP 分解。存在几种增量CP分解算法，但几乎所有这些算法都假设CP分解组件的数量在增量过程中保持固定。本文的主要贡献是研究了增量CP分解中如何添加分量。我们推导出相对于特殊基的增量 CP 分解的坐标表示，并显示张量等级的相关属性和这种增量 CP 分解的唯一性。在这种表示的框架下，可以通过使用交替最小化算法来解决所提出的方法。数值例子表明，与现有方法相比，所提出的算法在计算时间和数据拟合方面的良好性能。