One popular way to compute the CANDECOMP/PARAFAC (CP) decomposition of a tensor is to transform the problem into a sequence of overdetermined least squares subproblems with Khatri-Rao product (KRP) structure involving factor matrices. In this work, based on choosing the factor matrix randomly, we propose a mini-batch stochastic gradient descent method with importance sampling for those special least squares subproblems. Two different sampling strategies are provided. They can avoid forming the full KRP explicitly and computing the corresponding probabilities directly. The adaptive step size version of the method is also given. For the proposed method, we present its theoretical properties and comprehensive numerical performance. The results on synthetic and real data show that our method is effective and efficient, and for unevenly distributed data, it performs better than the corresponding one in the literature.

计算张量 CANDECOMP/PARAFAC (CP) 分解的一种流行方法是将问题转换为一系列具有涉及因子矩阵的 Khatri-Rao 乘积 (KRP) 结构的超定最小二乘子问题。在这项工作中，基于随机选择因子矩阵，我们针对那些特殊的最小二乘子问题提出了一种具有重要性采样的小批量随机梯度下降方法。提供了两种不同的采样策略。他们可以避免显式地形成完整的 KRP 并直接计算相应的概率。还给出了该方法的自适应步长版本。对于所提出的方法，我们介绍了其理论特性和综合数值性能。合成数据和真实数据的结果表明，我们的方法是有效和高效的，对于分布不均匀的数据，它的性能优于文献中的相应方法。