In this paper, we develop the notion of entropy for uniform hypergraphs via tensor theory. We employ the probability distribution of the generalized singular values, calculated from the higher-order singular value decomposition of the Laplacian tensors, to fit into the Shannon entropy formula. We show that this tensor entropy is an extension of von Neumann entropy for graphs. In addition, we establish results on the lower and upper bounds of the entropy and demonstrate that it is a measure of regularity for uniform hypergraphs in simulated and experimental data. We exploit the tensor train decomposition in computing the proposed tensor entropy efficiently. Finally, we introduce the notion of robustness for uniform hypergraphs.

中文翻译：

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均匀超图的张量熵

在本文中，我们通过张量理论发展了均匀超图的熵概念。我们采用广义奇异值的概率分布，从拉普拉斯张量的高阶奇异值分解计算，以适应香农熵公式。我们证明了这个张量熵是图的冯诺依曼熵的扩展。此外，我们建立了熵的下限和上限的结果，并证明它是模拟和实验数据中均匀超图的规律性度量。我们利用张量训练分解来有效地计算提出的张量熵。最后，我们引入了统一超图的鲁棒性概念。