Context-Aware Hypergraph Construction for Robust Spectral Clustering

Spectral clustering is a powerful tool for unsupervised data analysis. In this paper, we propose a context-aware hypergraph similarity measure (CAHSM), which leads to robust spectral clustering in the case of noisy data. We construct three types of hypergraph—the pairwise hypergraph, the k-nearest-neighbor (kNN) hypergraph, and the high-order over-clustering hypergraph. The pairwise hypergraph captures the pairwise similarity of data points; the kNN hypergraph captures the neighborhood of each point; and the clustering hypergraph encodes high-order contexts within the dataset. By combining the affinity information from these three hypergraphs, the CAHSM algorithm is able to explore the intrinsic topological information of the dataset. Therefore, data clustering using CAHSM tends to be more robust. Considering the intra-cluster compactness and the inter-cluster separability of vertices, we further design a discriminative hypergraph partitioning criterion (DHPC). Using both CAHSM and DHPC, a robust spectral clustering algorithm is developed. Theoretical analysis and experimental evaluation demonstrate the effectiveness and robustness of the proposed algorithm.

💡 Research Summary

The paper addresses several well‑known shortcomings of traditional spectral clustering—namely, difficulty in automatically determining the number of clusters, sensitivity to the choice of scaling parameters, vulnerability to noise and outliers, and limited ability to fuse heterogeneous information. To overcome these issues, the authors introduce a Context‑Aware Hypergraph Similarity Measure (CAHSM) that integrates three complementary hypergraph constructions: (1) a pairwise hypergraph that encodes conventional pairwise affinities; (2) a k‑nearest‑neighbor (kNN) hypergraph that captures local neighborhood relationships by forming a hyperedge for each data point together with its k nearest neighbors; and (3) an over‑clustering hypergraph that derives high‑order “contexts” from an over‑segmentation of the data, grouping vertices that share common properties into hyperedges.

Each hypergraph yields a similarity matrix: U for the pairwise hypergraph, B for the kNN hypergraph, and C for the over‑clustering hypergraph. The matrices are linearly combined with non‑negative weights (α, β, γ, α+β+γ = 1) to produce a final similarity matrix S = αU + βB + γC. This fusion allows the similarity measure to reflect both fine‑grained pairwise connections and higher‑order contextual information, making it more stable when individual vertices are corrupted by noise.

Beyond similarity construction, the authors propose a Discriminative Hypergraph Partitioning Criterion (DHPC) that simultaneously maximizes intra‑cluster compactness and inter‑cluster separability. DHPC is formulated as a trace‑ratio objective:

 max  Tr(Fᵀ L₁ F) / Tr(Fᵀ L₂ F)

where L₁ emphasizes connections within the same cluster and L₂ penalizes connections across clusters; F is the cluster indicator matrix. This objective can be relaxed to a generalized eigenvalue problem, enabling efficient computation. The trace‑ratio formulation also naturally yields a large eigen‑gap, facilitating automatic estimation of the number of clusters.

The overall algorithm proceeds as follows: (i) construct the three hypergraphs from the raw data and compute U, B, C; (ii) combine them into S; (iii) build a normalized Laplacian from S; (iv) solve the DHPC trace‑ratio problem to obtain a low‑dimensional embedding (the leading eigenvectors); (v) apply k‑means (or another simple clustering) to the embedding to produce the final cluster assignments.

Theoretical analysis demonstrates that the inclusion of high‑order contexts makes the similarity matrix robust to perturbations of individual points, while the DHPC objective enlarges the spectral gap, improving cluster separability and enabling cluster‑count inference.

Extensive experiments are conducted on synthetic datasets with varying noise levels, non‑convex shapes, and multi‑scale structures, as well as on real‑world benchmarks such as image segmentation (BSDS500), motion segmentation, and face image clustering. Quantitative metrics (NMI, accuracy, purity) show that the proposed method consistently outperforms baseline approaches, including Zelnik‑Manor’s local scaling, Noise‑Robust Spectral Clustering (NRSC), and earlier hypergraph‑based methods. Notably, when noise corrupts up to 30 % of the data, the performance degradation is minimal, whereas competing methods suffer substantial drops. Qualitative results illustrate that the over‑clustering hypergraph captures meaningful high‑order groupings, leading to cleaner segment boundaries and more coherent face clusters.

In summary, the paper makes three principal contributions: (1) a novel context‑aware hypergraph similarity measure that unifies pairwise, neighborhood, and high‑order grouping information; (2) a discriminative hypergraph partitioning criterion that jointly optimizes intra‑cluster cohesion and inter‑cluster separation; and (3) a complete spectral clustering framework that is robust to noise, can infer the number of clusters, and is applicable to a broad range of high‑dimensional, noisy data domains such as computer vision, bioinformatics, and text mining.