TT-LSQR for tensor least squares problems and application to data mining

We are interested in the numerical solution of the tensor least squares problem minX‖F-∑i=1ℓX×1A1(i)×2A2(i)⋯×dAd(i)‖F,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \min _{\mathcal {X}} \Vert \mathcal {F} - \sum _{i =1}^{\ell } \mathcal {X} \times _1 A_1^{(i)} \times _2 A_2^{(i)} \cdots \times _d A_d^{(i)} \Vert _F, $$\end{document}where X∈Rm1×m2×⋯×md\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}\in \mathbb {R}^{m_1 \times m_2 \times \cdots \times m_d}$$\end{document}, F∈Rn1×n2×⋯×nd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}\in \mathbb {R}^{n_1\times n_2 \times \cdots \times n_d}$$\end{document} are tensors with d dimensions, and the coefficients Aj(i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_j^{(i)}$$\end{document} 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 Aj(i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_j^{(i)}$$\end{document}. 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.