Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

A tensor rank theory and maximum full rank subtensors

Pages: 273 – 286



Liqun Qi (Dept. of Mathematics, Hangzhou Dianzi University, Hangzhou, China; Dept. of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong; and Future Network Theory Lab, Huawei Tech. Investment Co. Ltd., Shatin, N.T., Hong Kong)

Xinzhen Zhang (School of Mathematics, Tianjin University, Tianjin China)

Yannan Chen (School of Mathematical Sciences, South China Normal University, Guangzhou, China)


A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.


tensor rank axioms, full rank tensors, the max-full-rank-subtensor property, the max-Tucker rank, the submax-Tucker rank

2010 Mathematics Subject Classification

15A69, 15A83

The work of Xinzhen Zhang was supported by NSFC (Grant No. 11871369).

The work of Yannan Zhang was supported by the National Natural Science Foundation of China (11771405), and by the Guangdong Basic and Applied Basic Research Foundation (2020A1515010489).

Received 20 May 2020

Accepted 6 September 2020

Published 24 March 2021