Fuzzy Hypergraph Clustering via Higher-Order Modularity Maximization

Conventional clustering methods mainly characterize cluster structures using geometric proximity or pairwise similarities, which may be insufficient for data with complex higher-order dependencies. This paper proposes a Fuzzy Hypergraph Clustering (FHC) method that transforms ordinary feature data into a data-induced uniform hypergraph and performs clustering through higher-order modularity maximization. Specifically, local neighborhoods are first encoded as weighted hyperedges to represent multi-sample relations. A degree-corrected null model is then constructed, based on which fuzzy hypergraph modularity is defined to measure the excess concentration of observed higher-order relations within fuzzy clusters relative to random expectation. The proposed objective reduces to fuzzy graph modularity when the hyperedge order is two and, at unit resolution, further reduces to classical Newman–Girvan modularity under hard assignments. To optimize the resulting nonconvex objective, a simplex-constrained projected gradient ascent algorithm with Armijo backtracking is developed. The optimization is implemented directly over hyperedges without explicitly constructing the high-order adjacency tensor. Experiments on synthetic data and thirteen benchmark datasets show that FHC achieves the strongest overall ranking among the compared methods while maintaining stable performance over a broad parameter range and competitive computational cost. On the Alzheimer’s MRI dataset, FHC obtains the best NMI and ARI and the second-best ACC, further supporting the effectiveness of higher-order fuzzy modularity for clustering feature data with complex relational structures.