Logarithmic Schatten-$p$p Norm Minimization for Tensorial Multi-View Subspace Clustering

The low-rank tensor could characterize inner structure and explore high-order correlation among multi-view representations, which has been widely used in multi-view clustering. Existing approaches adopt the tensor nuclear norm (TNN) as a convex approximation of non-convex tensor rank function. However, TNN treats the different singular values equally and over-penalizes the main rank components, leading to sub-optimal tensor representation. In this paper, we devise a better surrogate of tensor rank, namely the tensor logarithmic Schatten-$p$ norm ($\text{TLS}_{p}$N), which fully considers the physical difference between singular values by the non-convex and non-linear penalty function. Further, a tensor logarithmic Schatten-$p$ norm minimization ($\text{TLS}_{p}$NM)-based multi-view subspace clustering ($\text{TLS}_{p}$NM-MSC) model is proposed. Specially, the proposed $\text{TLS}_{p}$NM can not only protect the larger singular values encoded with useful structural information, but also remove the smaller ones encoded with redundant information. Thus, the learned tensor representation with compact low-rank structure will well explore the complementary information and accurately characterize the high-order correlation among multi-views. The alternating direction method of multipliers (ADMM) is used to solve the non-convex multi-block $\text{TLS}_{p}$NM-MSC model where the challenging $\text{TLS}_{p}$NM problem is carefully handled. Importantly, the algorithm convergence analysis is mathematically established by showing that the sequence generated by the algorithm is of Cauchy and converges to a Karush-Kuhn-Tucker (KKT) point. Experimental results on nine benchmark databases reveal the superiority of the $\text{TLS}_{p}$NM-MSC model.