论文标题
实用的最小二乘张量环分解
Practical Alternating Least Squares for Tensor Ring Decomposition
论文作者
论文摘要
张量环（TR）分解已被广泛用作各种应用中的有效方法，以发现多维数据中隐藏的低级别模式。 TR分解的一种众所周知的方法是交替的最小二乘（ALS）。但是，它通常遭受臭名昭著的中间数据爆炸问题，尤其是对于大型张量。在本文中，我们提供了解决此问题并设计三种基于ALS的算法的两种策略。具体而言，第一种策略用于简化ALS子问题的正常方程式的系数矩阵的计算，该策略可以充分利用子问题的系数矩阵的结构，因此使相应的算法在计算时间方面的表现胜于常规ALS方法。第二种策略是通过QR因子在TR核上稳定ALS子问题，因此，与我们的第一个算法相比，相应的算法在数值上更稳定。提供有关合成和实际数据的广泛数值实验，以说明和确认上述结果。此外，我们还提出了所提出的算法的复杂性分析。
Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low-rank patterns in multidimensional data. A well-known method for TR decomposition is the alternating least squares (ALS). However, it often suffers from the notorious intermediate data explosion issue, especially for large-scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS-based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for the ALS subproblems, which takes full advantage of the structure of the coefficient matrices of the subproblems and hence makes the corresponding algorithm perform much better than the regular ALS method in terms of computing time. The second strategy is to stabilize the ALS subproblems by QR factorizations on TR-cores, and hence the corresponding algorithms are more numerically stable compared with our first algorithm. Extensive numerical experiments on synthetic and real data are given to illustrate and confirm the above results. In addition, we also present the complexity analyses of the proposed algorithms.
