摘要
Based on the structure of the rank-1 matrix and the different unfolding ways of the tensor, we present two types of structured tensors which contain the rank-1 tensors as special cases. We study some properties of the ranks and the best rank-r approximations of the structured tensors. By using the upper-semicontinuity of the matrix rank, we show that for the structured tensors, there always exist the best rank-r approximations. This can help one to better understand the sequential unfolding singular value decomposition （SVD） method for tensors proposed by J. Salmi et al. [IEEE Trans Signal Process, 2009, 57（12）： 4719-4733] and offer a generalized way of low rank approximations of tensors. Moreover, we apply the structured tensors to estimate the upper and lower bounds of the best rank-1 approximations of the 3rd-order and 4th-order tensors, and to distinguish the well written and non-well written digits.
Based on the structure of the rank-1 matrix and the different unfolding ways of the tensor, we present two types of structured tensors which contain the rank-1 tensors as special cases. We study some properties of the ranks and the best rank-r approximations of the structured tensors. By using the upper-semicontinuity of the matrix rank, we show that for the structured tensors, there always exist the best rank-r approximations. This can help one to better understand the sequential unfolding singular value decomposition （SVD） method for tensors proposed by J. Salmi et al. [IEEE Trans Signal Process, 2009, 57（12）： 4719-4733] and offer a generalized way of low rank approximations of tensors. Moreover, we apply the structured tensors to estimate the upper and lower bounds of the best rank-1 approximations of the 3rd-order and 4th-order tensors, and to distinguish the well written and non-well written digits.
基金
Acknowledgements The authors would like to thank the anonymous referees for their valuable suggestions and helpful comments, which greatly improved the content and presentation of the paper. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11071192, 11171270, 11201362）, the International Science and Technology Cooperation Program of China （Grant No. 2010DFA14700）, and the Fundamental Research Funds for the Central Universities.
