Kernel fuzzy c-means (KFCM) is a significantly improved version of fuzzy c-means (FCM) for processing linearly inseparable datasets. However, for fuzzification parameter $m=1$, the problem of KFCM (kernel fuzzy c-means) cannot be solved by Lagrangian optimization. To solve this problem, an equivalent model, called kernel probabilistic k-means (KPKM), is proposed here. The novel model relates KFCM to kernel k-means (KKM) in a unified mathematic framework. Moreover, the proposed KPKM can be addressed by the active gradient projection (AGP) method, which is a nonlinear programming technique with constraints of linear equalities and linear inequalities. To accelerate the AGP method, a fast AGP (FAGP) algorithm was designed. The proposed FAGP uses a maximum-step strategy to estimate the step length, and uses an iterative method to update the projection matrix. Experiments demonstrated the effectiveness of the proposed method through a performance comparison of KPKM with KFCM, KKM, FCM and k-means. Experiments showed that the proposed KPKM is able to find nonlinearly separable structures in synthetic datasets. Ten real UCI datasets were used in this study, and KPKM had better clustering performance on at least six datsets. The proposed fast AGP requires less running time than the original AGP, and it reduced running time by 76–95% on real datasets.

Contrastive-based clustering models usually rely on a large number of negative pairs to capture uniform representations, which requires a large batch size and high computational complexity. In contrast, some self-supervised methods perform non-contrastive learning to capture discriminative representations only with positive pairs, but suffer from the collapse of clustering. To solve these issues, a novel end-to-end self-supervised clustering model is proposed in this paper. The basic self-supervised learning network is first modified, followed by the incorporation of a Softmax layer to obtain cluster assignments as data representation. Then, adversarial learning on the cluster assignments is integrated into the methods to further enhance discrimination across different clusters and mitigate the collapse between clusters. To further encourage clustering-oriented guidance, a new cluster-level discrimination is assembled to promote clustering performance by measuring the self-correlation between the learned cluster assignments. Experimental results on real-world datasets exhibit better performance of the proposed model compared with the existing deep clustering methods.

Clustering nonlinearly separable datasets is always an important problem in unsupervised machine learning. Graph cut models provide good clustering results for nonlinearly separable datasets, but solving graph cut models is an NP hard problem. A novel graph-based clustering algorithm is proposed for nonlinearly separable datasets. The proposed method solves the min cut model by iteratively computing only one simple formula. Experimental results on synthetic and benchmark datasets indicate the potential of the proposed method, which is able to cluster nonlinearly separable datasets with less running time.

K-means is a classical clustering algorithm with wide applications. However, soft K-means, or fuzzy c-means at m=1, remains unsolved since 1981. To address this challenging open problem, we propose a novel clustering model, i.e. Probabilistic K-Means (PKM), which is also a nonlinear programming model constrained on linear equalities and linear inequalities. In theory, we can solve the model by active gradient projection, while inefficiently. Thus, we further propose maximum-step active gradient projection and fast maximum-step active gradient projection to solve it more efficiently. By experiments, we evaluate the performance of PKM and how well the proposed methods solve it in five aspects: initialization robustness, clustering performance, descending stability, iteration number, and convergence speed.