Multi-view Spectral Clustering on the Grassmannian Manifold With Hypergraph Representation

TL;DR

This work tackles the limitations of pairwise graphs in multi-view clustering by introducing a hypergraph-based framework that learns per-view hypergraphs from sparse representations and fuses them via an optimization on the Grassmannian $\mathcal{G}(k,n)$. The MHSCG method constructs hypergraph Laplacians for each view, jointly optimizes the top-$k$ eigen-subspaces with a consistency penalty against a consensus view, and solves the constrained problem using alternating Riemannian optimization with adaptive $\lambda_l$. Key contributions include per-view hypergraph generation from sparse coefficients, a multi-view objective that balances spectral clustering quality with cross-view alignment, and an unconstrained Grassmannian formulation that mitigates local maxima and reduces dimensionality. Experimental results on four multi-view datasets demonstrate state-of-the-art clustering performance and robustness to initialization, validating the practical impact of combining sparse hypergraph representations with Grassmannian optimization for multi-view learning.

Abstract

Graph-based multi-view spectral clustering methods have achieved notable progress recently, yet they often fall short in either oversimplifying pairwise relationships or struggling with inefficient spectral decompositions in high-dimensional Euclidean spaces. In this paper, we introduce a novel approach that begins to generate hypergraphs by leveraging sparse representation learning from data points. Based on the generated hypergraph, we propose an optimization function with orthogonality constraints for multi-view hypergraph spectral clustering, which incorporates spectral clustering for each view and ensures consistency across different views. In Euclidean space, solving the orthogonality-constrained optimization problem may yield local maxima and approximation errors. Innovately, we transform this problem into an unconstrained form on the Grassmannian manifold. Finally, we devise an alternating iterative Riemannian optimization algorithm to solve the problem. To validate the effectiveness of the proposed algorithm, we test it on four real-world multi-view datasets and compare its performance with seven state-of-the-art multi-view clustering algorithms. The experimental results demonstrate that our method outperforms the baselines in terms of clustering performance due to its superior low-dimensional and resilient feature representation.

Figures (4)

• Figure 1: Schematic diagram of the multi-view hypergraph spectral clustering on the Grassmannian manifold. $\mathbf{\Theta}^{(l)}$ denotes the hypergraph laplacian variant matrix of $l$-th view, and the initialized eigenvector matrix $\mathbf{F}^{(l)}$ can be obtained through its spectral decomposition, where $l=1,\ldots,r$. $[\mathbf{F}^{(l)}]$ denote the equivalence classes of $\mathbf{F}^{(l)}$. Each point on manifold $\mathcal{G}(k,n)$ corresponds to a set of k-dimensional dominant subspaces of the eigenspace.
• Figure 2: Clustering accuracy of MHSCG with different initial values of $\lambda_l$. (a) 3sources. (b) BBC. (c) Mfeat Digits. (d) MSRCv1.
• Figure 3: Convergence curves of MHSCG. (a) 3sources. (b) BBC. (c) Mfeat Digits. (d) MSRCv1.
• Figure 4: Comparison of MHSCG with seven baselines.

Theorems & Definitions (3)

• Lemma 1
• Remark 1
• Remark 2