论文标题
张量排名理论和最大全等级子镜
A Tensor Rank Theory and Maximum Full Rank Subtensors
论文作者
论文摘要
矩阵始终具有完整的排名子序列，使得该矩阵的排名等于该子序列的等级。该属性是矩阵等级理论的角石之一。我们将此属性称为最大级别submatrix属性。张量排名在低等级张量近似，张量完成和张量恢复中起着至关重要的作用。但是，他们的理论尚未成熟。我们可以为张量排列设置公理系统吗？我们可以将最大级别含量的属性扩展到张量吗？我们在本文中探讨了这些。我们首先提出一些张量级函数的公理。然后，我们引入适当的张量排名函数。 CP等级是张量排名函数，但不合适。有两个适当的张量排名函数，最大塔克等级和submax-tucker等级与塔克分解相关。我们在张量等级函数之间定义了部分顺序，并表明存在独特的最小张量级函数。我们介绍了完整的等级张量概念，并定义了最大额定量的属性。我们显示了最大 - 塔克张量级函数，最小的张量级函数具有此属性。我们定义了任意适当的张量级函数的关闭，并表明它仍然是适当的张量级函数，并且具有最大额定功能。还提出了超级塔克等级的应用。
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.
