关于连接张量结构及其在张量特征值问题中的应用
On the structure of join tensors with applications to tensor eigenvalue problems
AI总结：
研究连接张量的结构，推导其多项式分解和张量列车分解，并探讨其存储复杂度及特征值上界。
摘要：
我们研究了连接张量的结构，将其视为多变量扩展的格论连接矩阵。在一般的连接半格上，推导了连接张量的多项式分解（即秩-1张量的线性组合）和张量列车分解的显式公式。我们讨论了所获得的分解在秩上的最优条件，并通过数值方法检验了所得分解对一类LCM张量的存储复杂度。此外，我们还通过数值方法检验了LCM张量张量特征值的理论上界。
英文摘要：
We investigate the structure of join tensors, which may be regarded as the multivariable extension of lattice-theoretic join matrices. Explicit formulae for a polyadic decomposition (i.e., a linear combination of rank-1 tensors) and a tensor-train decomposition of join tensors are derived on general join semilattices. We discuss conditions under which the obtained decompositions are optimal in rank, and examine numerically the storage complexity of the obtained decompositions for a class of LCM tensors as a special case of join tensors. In addition, we investigate numerically the sharpness of a theoretical upper bound on the tensor eigenvalues of LCM tensors.
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