Title: New Ranks for Even-Order Tensors and Their Applications in Low-Rank Tensor Optimization
Abstract:In this paper, we propose three new notions of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank. We discuss the bounds between these new tens...In this paper, we propose three new notions of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank. We discuss the bounds between these new tensor ranks and the CP(CANDECOMP/PARAFAC)-rank and the symmetric CP-rank of an even-order tensor. In particular, we show: (1) these newly defined ranks actually coincide with each other if the even-order tensor in question is super-symmetric; (2) the CP-rank and symmetric CP-rank for a fourth-order tensor can be both lower and upper bounded (up to a constant factor) by the corresponding M-rank, and the bounds are tight for asymmetric tensors. In addition, we manage to identify a class of tensors whose CP-rank can be easily determined, and the CP-rank of the tensor equals the M-rank but is strictly larger than the Tucker rank. Since the M-rank is much easier to compute than the CP-rank, we propose to replace the CP-rank by the M-rank in the low-CP-rank tensor recovery model. Numerical results suggest that when the CP-rank is not very small our method outperforms the low-n-rank approach which is a currently popular model in low-rank tensor recovery. This shows that the M-rank is indeed an effective and easily computable approximation of the CP-rank.Read More
Publication Year: 2015
Publication Date: 2015-01-15
Language: en
Type: preprint
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Cited By Count: 1
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