A Tensor Rank Theory and The Sub-Full-Rank Property
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One fundamental property in matrix theory is that the rank of a matrix is always equal to the maximum value of all of its full rank submatrices. We call this property the sub-full-rank property. Matrix datasets are in general not of full rank. But we may always identify their full rank submatrices with maximum rank values. In this paper, we explore this property for tensors. We first present a theory for tensor ranks such that they are natural extension of matrix ranks. We present some axioms for tensor rank functions. Then we introduce strongly proper tensor rank functions. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We show that the smallest tensor rank function is strongly proper and has the sub-full-rank property. We also show that the closure of a strongly proper tensor rank function is a strongly proper tensor rank function with the sub-full-rank property. An example of a strongly proper tensor rank function, which is easily computable, is the submax-Tucker rank function, which is associated with the Tucker decomposition.
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