While multilinear algebra appears natural for studying the multiway interactions modeled by hypergraphs, tensor methods for general hypergraphs have been stymied by theoretical and practical barriers. A recently proposed adjacency tensor is applicable to nonuniform hypergraphs, but is prohibitively costly to form and analyze in practice. We develop tensor times same vector (TTSV) algorithms for this tensor which improve complexity from $O(n^r)$ to a low-degree polynomial in $r$, where $n$ is the number of vertices and $r$ is the maximum hyperedge size. Our algorithms are implicit, avoiding formation of the order $r$ adjacency tensor. We demonstrate the flexibility and utility of our approach in practice by developing tensor-based hypergraph centrality and clustering algorithms. We also show these tensor measures offer complementary information to analogous graph-reduction approaches on data, and are also able to detect higher-order structure that many existing matrix-based approaches provably cannot.

翻译：尽管多线性代数自然适用于研究超图建模的多路相互作用，但张量方法在通用超图中的应用一直受到理论和实践障碍的阻碍。近期提出的邻接张量虽然适用于非均匀超图，但在实践中其构建和分析成本过高。我们针对该张量开发了张量同向量乘积（TTSV）算法，将计算复杂度从$O(n^r)$降低至$r$的低次多项式，其中$n$为顶点数，$r$为最大超边规模。我们的算法采用隐式方式，避免构建$r$阶邻接张量。通过开发基于张量的超图中心性和聚类算法，我们在实践中展示了该方法的灵活性与实用性。同时证明，这些张量度量不仅能提供与数据上同类图约简方法互补的信息，还能检测许多现有基于矩阵的方法无法证明的高阶结构。