用于张量最小二乘问题及数据挖掘应用的TT-LSQR
TT-LSQR For Tensor Least Squares Problems and Application to Data Mining *
AI总结：
本文提出基于张量链表示的TT-LSQR方法求解张量最小二乘问题，并结合随机投影降低计算成本，成功应用于信息检索中的文档分类。
摘要：
我们关注张量最小二乘问题的数值解：min_X || F - sum_{i=1}^l X x_1 A_1^(i) x_2 A_2^(i) ... x_d A_d^(i) ||_F，其中X和F是d维张量，系数A_j^(i)为尺寸相符的高矩阵。我们首先以张量链表示为关键要素，描述了Paige和Saunders经典LSQR方法的张量实现。我们还展示了如何结合随机投影以降低处理高矩阵的计算成本。随后，我们将该方法应用于信息检索中的问题，即根据给定关键词将新查询文档在已分类文档中进行归类。
英文摘要：
We are interested in the numerical solution of the tensor least squares problem \[ \min_{\mathcal{X}} \| \mathcal{F} - \sum_{i =1}^{\ell} \mathcal{X} \times_1 A_1^{(i)} \times_2 A_2^{(i)} \cdots \times_d A_d^{(i)} \|_F, \] where $\mathcal{X}\in\mathbb{R}^{m_1 \times m_2 \times \cdots \times m_d}$, $\mathcal{F}\in\mathbb{R}^{n_1\times n_2 \times \cdots \times n_d}$ are tensors with $d$ dimensions, and the coefficients $A_j^{(i)}$ are tall matrices of conforming dimensions. We first describe a tensor implementation of the classical LSQR method by Paige and Saunders, using the tensor-train representation as key ingredient. We also show how to incorporate sketching to lower the computational cost of dealing with the tall matrices $A_j^{(i)}$. We then use this methodology to address a problem in information retrieval, the classification of a new query document among already categorized documents, according to given keywords.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
