### Abstract

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors {x_{(i)}} with x_{(i)}∈R^{n1×⋯×nd } in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram-Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α_{i} in an arbitrary tensor v=∑_{i}α_{i}x_{(i)}. The orthonormal Q factor in the QR decomposition X≡[x_{(1)},⋯,x_{(p)}]=QR cannot be computed but expressed as XR^{-1} when required. The resulting algorithm has an O(p^{2}dn) computational complexity, with n=maxn_{i}. Some illustrative examples in the numerical solution of tensor linear equations are presented.

Original language | English |
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Pages (from-to) | 292-316 |

Number of pages | 25 |

Journal | Linear Algebra and Its Applications |

Volume | 491 |

DOIs | |

Publication status | Published - 15 Feb 2016 |

### Keywords

- Column pivoting
- Gram-Schmidt orthogonalization
- Kronecker product
- Large-scale problem
- Linear equation
- Low-rank representation
- Multilinear algebra
- QR decomposition

## Cite this

*Linear Algebra and Its Applications*,

*491*, 292-316. https://doi.org/10.1016/j.laa.2015.08.035