Proximal Plane Clustering via Eigenvalues

Multiple Flat Projections for Cross-Manifold Clustering

Cross-manifold clustering is an extreme challenge learning problem. Since the low-density hypothesis is not satisfied in cross-manifold problems, many traditional clustering methods failed to discover the cross-manifold structures. In this article, we propose multiple flat projections clustering (MFPC) for cross-manifold clustering. In our MFPC, the given samples are projected into multiple localized flats to discover the global structures of implicit manifolds. Thus, the intersected clusters are distinguished in various projection flats. In MFPC, a series of nonconvex matrix optimization problems is solved by a proposed recursive algorithm. Furthermore, a nonlinear version of MFPC is extended via kernel tricks to deal with a more complex cross-manifold learning situation. The synthetic tests show that our MFPC works on the cross-manifold structures well. Moreover, experimental results on the benchmark datasets and object tracking videos show excellent performance of our MFPC compared with some state-of-the-art manifold clustering methods.

General Plane-Based Clustering With Distribution Loss

In this article, we propose a general model for plane-based clustering. The general model reveals the relationship between cluster assignment and cluster updating during clustering implementation, and it contains many existing plane-based clustering methods, e.g., k-plane clustering, proximal plane clustering, twin support vector clustering, and their extensions. Under this general model, one may obtain an appropriate clustering method for a specific purpose. The general model is a procedure corresponding to an optimization problem, which minimizes the total loss of the samples. Thereinto, the loss of a sample derives from both within-cluster and between-cluster information. We discuss the theoretical termination conditions and prove that the general model terminates in a finite number of steps at a local or weak local solution. Furthermore, we propose a distribution loss function that fluctuates with the input data and introduce it into the general model to obtain a plane-based clustering method (DPC). DPC can capture the data distribution precisely because of its statistical characteristics, and its termination that finitely terminates at a weak local solution is given immediately based on the general model. The experimental results show that our DPC outperforms the state-of-the-art plane-based clustering methods on many synthetic and benchmark data sets.

L1-Norm Distance Minimization-Based Fast Robust Twin Support Vector $k$ -Plane Clustering

Twin support vector clustering (TWSVC) is a recently proposed powerful k-plane clustering method. It, however, is prone to outliers due to the utilization of squared L2-norm distance. Besides, TWSVC is computationally expensive, attributing to the need of solving a series of constrained quadratic programming problems (CQPPs) in learning each clustering plane. To address these problems, this brief first develops a new k-plane clustering method called L1-norm distance minimization-based robust TWSVC by using robust L1-norm distance. To achieve this objective, we propose a novel iterative algorithm. In each iteration of the algorithm, one CQPP is solved. To speed up the computation of TWSVC and simultaneously inherit the merit of robustness, we further propose Fast RTWSVC and design an effective iterative algorithm to optimize it. Only a system of linear equations needs to be computed in each iteration. These characteristics make our methods more powerful and efficient than TWSVC. We also conduct some insightful analysis on the existence of local minimum and the convergence of the proposed algorithms. Theoretical insights and effectiveness of our methods are further supported by promising experimental results.

Twin support vector machine for clustering

The twin support vector machine (TWSVM) is one of the powerful classification methods. In this brief, a TWSVM-type clustering method, called twin support vector clustering (TWSVC), is proposed. Our TWSVC includes both linear and nonlinear versions. It determines k cluster center planes by solving a series of quadratic programming problems. To make TWSVC more efficient and stable, an initialization algorithm based on the nearest neighbor graph is also suggested. The experimental results on several benchmark data sets have shown a comparable performance of our TWSVC.