Sparse-Gradient-Regularized Multi-View Clustering with Joint ℓ2,log Sparsity and a Tensor γ* Rank Surrogate

Multi-view clustering seeks a consensus partition from heterogeneous feature views while retaining view-specific information. We propose SGLog-γ∗-MSC, a nonconvex multi-view subspace clustering framework that combines graph total variation, an ℓ2,log penalty, and a tensor γ∗ spectral penalty. These terms promote locally consistent self-representations, model sample-wise corruption, and capture shared low-rank structure across views, respectively. We derive an alternating augmented-Lagrangian algorithm with candidate-selection rules that return global minimizers for both nonconvex proximal subproblems. We also state explicit conditions under which accumulation points satisfy the KKT system and establish conditional whole-sequence convergence through the Kurdyka–Łojasiewicz framework. Hyperparameters are selected by a label-free protocol specified before the audited rerun; ground-truth labels are never used during selection. Across 20 recorded k-means++ initializations of each fixed embedding, the method attains NMI 0.8206±0.0155 on Yale and 0.8962±0.0211 on Scene-15. Relative to nine literature-reported baselines, these are the highest reported NMI values on the two datasets. On UCI digits, BBCSport, and ORL, the method reaches NMI 0.9837, 0.9634, and 0.9892, respectively, ranking second only to HLR-M2VS in the descriptive cross-paper comparison. Component-wise ablation identifies the ℓ2,log term as the largest and most consistent contributor, while the effects of the γ∗ surrogate and sparse-gradient term depend on the dataset. Compared with the tensor nuclear norm, the γ∗ surrogate improves performance on BBCSport, ORL, and UCI digits; sensitivity analysis shows that an interior γ also improves performance on Scene-15. Together, these results support combining robust error modeling with local and shared structural regularization while emphasizing the dataset dependence of individual components.