Graph-based Clustering Revisited: A Relaxation of Kernel $k$-Means Perspective

The well-known graph-based clustering methods, including spectral clustering, symmetric non-negative matrix factorization, and doubly stochastic normalization, can be viewed as relaxations of the kernel $k$-means approach. However, we posit that these methods excessively relax their inherent low-rank, nonnegative, doubly stochastic, and orthonormal constraints to ensure numerical feasibility, potentially limiting their clustering efficacy. In this paper, guided by our theoretical analyses, we propose \textbf{Lo}w-\textbf{R}ank \textbf{D}oubly stochastic clustering (\textbf{LoRD}), a model that only relaxes the orthonormal constraint to derive a probabilistic clustering results. Furthermore, we theoretically establish the equivalence between orthogonality and block diagonality under the doubly stochastic constraint. By integrating \textbf{B}lock diagonal regularization into LoRD, expressed as the maximization of the Frobenius norm, we propose \textbf{B-LoRD}, which further enhances the clustering performance. To ensure numerical solvability, we transform the non-convex doubly stochastic constraint into a linear convex constraint through the introduction of a class probability parameter. We further theoretically demonstrate the gradient Lipschitz continuity of our LoRD and B-LoRD enables the proposal of a globally convergent projected gradient descent algorithm for their optimization. Extensive experiments validate the effectiveness of our approaches. The code is publicly available at https://github.com/lwl-learning/LoRD.