## Abstract

Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f. When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f. The critical rank-one tensors for f lie in a linear subspace H_{f}, the critical space of f. Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space H_{f}. This is the part of Eckart–Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space H_{f} is spanned by the complex critical rank-one tensors. Since f itself belongs to H_{f}, we deduce that also f itself is a linear combination of its critical rank-one tensors.

Original language | English |
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Article number | 27 |

Number of pages | 13 |

Journal | Research in the Mathematical Sciences |

Volume | 5 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jun 2018 |

## Keywords

- Best rank-k approximation
- Eckart–Young Theorem
- Tensor
- Eckart-Young Theorem