Graph clustering technique is highly effective in detecting complex-shaped clusters, in which graph building is a crucial step. Nevertheless, building a reasonable graph that can exhibit high connectivity within clusters and low connectivity across clusters is challenging. Herein, we design a max-ascent-angle graph called the “Y-graph”, a high-sparse graph that automatically allocates dense edges within clusters and sparse edges across clusters, regardless of their shapes or dimensionality. In the graph, every point$x$is allowed to connect its nearest higher-density neighbor$\delta$, and another higher-density neighbor$\gamma$, satisfying that the angle$\angle \delta x\gamma$is the largest, called “max-ascent-angle”. By seeking the max-ascent-angle, points are automatically connected as the Y-graph, which is a reasonable graph that can effectively balance inter-cluster connectivity and intra-cluster non-connectivity. Besides, an edge weight function is designed to capture the similarity of the neighbor probability distribution, which effectively represents the density connectivity between points. By employing the Normalized-Cut (Ncut) technique, a Ncut-Y algorithm is proposed. Benefiting from the excellent performance of Y-graph, Ncut-Y can fast seek and cut the edges located in the low-density boundaries between clusters, thereby, capturing clusters effectively. Experimental results on both synthetic and real datasets demonstrate the effectiveness of Y-graph and Ncut-Y.

Current methods for achieving fixed-time convergence in continuous sliding mode control are extensively studied, whereas there are limited studies on fixed-step convergence in discrete sliding mode control. The control objective of this paper aims to develop control schemes assuring that the number of convergence steps of the tracking error remains within the desired range, even as the initial value increases, and this paper proposes a guaranteed fixed-step convergence approach to form the reaching law. An upper bound on convergent steps, independent of the initial value, is established, thereby having the fixed-step convergence property . The performance assessment, including attractiveness, invariance, and convergence steps, is provided to meet the required specifications. The steady-state band results offer guidance in selecting parameters to achieve the control objective. To achieve fixed-step convergence of the tracking error, one can develop a dead-beat terminal sliding mode control scheme, where the tracking error is governed by the prescribed error dynamics. To illustrate the approach more concretely, specific designs are proposed for both switching and non-switching reaching laws aimed at ensuring fixed-step convergence. The simulation and experiment results validate the performance evaluation and demonstrate effectiveness of the proposed control schemes.

Most existing clustering algorithms require presetting cluster number and often fail to capture complex shapes. Herein, we propose a clustering algorithm by pruning a density-boosting cluster tree of density mounts—DEnsity MOuntains Separation clustering algorithm (DEMOS). A cluster is assumed to be a density-connected area with multiple (or a single) density mounts (i.e., single-peak clusters) and a relatively large dis-connectivity from density-connected areas of higher densities. Based on this assumption, DEMOS can easily detect the number of clusters and robustly reconstruct their complex shapes. It first builds the dataset into a peak graph, where each density peak represents a density mount. A multi-valley-link-based connectivity estimation method is embedded to efficiently estimate the connectivity between density peaks during peak graph building. Then, by applying a new linkage metric designed based on our assumption, DEMOS builds density mounts into a reasonably density-boosting cluster tree. After obtaining a robust center detection in a clarity-enhancing decision graph (i.e., a two-dimensional plot for detecting centers), DEMOS prunes the cluster tree into final clusters to finish clustering. Experimental results on both synthetic and real datasets demonstrated the effectiveness of DEMOS and its applicability to large-scale data clustering.