The affinity propagation (AP) clustering algorithm has received a lot of attention over the past few years. AP is efficient and insensitive to initialization, and generates clustering results with lower error and in less time. However, there are still two key limitations: AP-related algorithms cannot identify outliers in clusters. And they are usually not ideal for processing nonlinear data. To address the above issues, we propose a geodesic affinity propagation clustering algorithm based on angle-based outlier factor (ABOF-GAP). First, outliers are identified according to the value of angle-based outlier factor. Besides, Euclidean distance is replaced with geodesic distance to measure similarity. Experiments on synthetic data and real data illustrate the effectiveness of the ABOF-GAP algorithm.INDEX TERMS Affinity propagation (AP), geodesic distances, outlier identification, angle-based outlier factor (ABOF).

In this paper, a new preconditioning method is proposed for the linear system arising from the elliptic optimal control problem. It is based on row permutations of the linear system and approximations of the corresponding Schur complement inspired by the matching strategy. The eigenvalue bounds of the preconditioned matrices are shown to be independent of mesh size and regularization parameter. Numerical results illustrate the efficiency of the proposed preconditioning methods.