A Unified Theory for Tensor Ranks and its Application
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In this paper, we present a unified theory for tensor ranks such that they are natural extension of matrix ranks. We present some axioms for tensor rank functions. The CP rank, the max-Tucker rank and the submax-Tucker rank are tensor rank functions. The CP rank is subadditive but not proper. The max-Tucker rank naturally arises from the Tucker decomposition. It is proper and subadditive, but not strongly proper. The submax-Tucker rank is also associated with the Tucker decomposition, but is a new tensor rank function. It is strongly proper but not subadditive. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. The CP rank, and the max-Tucker rank are not the smallest tensor rank function. We define the closure of a strongly proper tensor rank function, and show that it is also a strongly proper tensor rank function. A strongly proper tensor rank function is closed if it is equal to its closure. We show that the smallest tensor rank function is strongly proper and closed. Our theoretic analysis indicates that the submax-Tucker rank is a good choice for low rank tensor approximation and tensor completion. An application of the submax-Tucker rank is presented.
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