#### A Tensor Rank Theory, Full Rank Tensors and The Sub-Full-Rank Property

##### Liqun Qi, Xinzhen Zhang, Yannan Chen

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. We call this property the sub-full-rank property. This property is one of the corner stones of the matrix rank theory. 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 full rank concept and the sub-full-rank 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 associated with the Tucker decomposition, the max-Tucker rank and the submax-Tucker rank. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the concept of full rank tensors, and show that the max-Tucker tensor rank function and the smallest tensor rank function have the sub-full-rank property. We define the closure for a proper tensor rank function, and show that the closure of any proper tensor rank function is a proper tensor rank function with the sub-full-rank property. An application of the submax-Tucker rank is also presented.

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