We are interested in the numerical solution of the tensor least squares problem \( \min _{\mathcal {X}} \Vert \mathcal {F} - \sum _{i =1}^{\ell } \mathcal {X} \times _1 A_1^{(i)} \times _2 A_2^{(i)} \cdots \times _d A_d^{(i)} \Vert _F, \) 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.
