Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The persistence of outliers and clusters of arbitrary shape is inferred from the resulting hierarchy. We demonstrate the potential of the algorithm on selected datasets in which outliers play relevant roles, consisting of images, medical and economic data. These methods can provide meaningful clusters in situations in which other techniques fail to do so.

Federated learning is a learning paradigm to enable collaborative learning across different parties without revealing raw data. Notably, vertical federated learning (VFL), where parties share the same set of samples but only hold partial features, has a wide range of real-world applications.

The growing adoption of Industrial Internet of Things (IIoT) technologies enables automated, real-time collection of manufacturing process data, unlocking new opportunities for data-driven product development. Current data-driven methods are generally applied within specific domains, such as design or manufacturing, with limited exploration of integrating design features and manufacturing process data. Since design decisions significantly affect manufacturing outcomes, such as error rates, energy consumption, and processing times, the lack of such integration restricts the potential for data-driven product design improvements. This paper presents a data-driven approach to mapping and analyzing the relationship between design features and manufacturing process data. A comprehensive system architecture is developed to ensure continuous data collection and integration. The linkage between design features and manufacturing process data serves as the basis for developing a machine learning model that enables automated design improvement suggestions. By integrating manufacturing process data with sustainability metrics, this approach opens new possibilities for sustainable product development.

Hierarchical clustering seeks to uncover nested structures in data by constructing a tree of clusters, where deeper levels reveal finer-grained relationships. Traditional methods, including linkage approaches, face three major limitations: (i) they always return a hierarchy, even if none exists, (ii) they are restricted to binary trees, even if the true hierarchy is non-binary, and (iii) they are highly sensitive to the choice of linkage function. In this paper, we address these issues by introducing the notion of a valid hierarchy and defining a partial order over the set of valid hierarchies. We prove the existence of a finest valid hierarchy, that is, the hierarchy that encodes the maximum information consistent with the similarity structure of the data set. In particular, the finest valid hierarchy is not constrained to binary structures and, when no hierarchical relationships exist, collapses to a star tree. We propose a simple two-step algorithm that first constructs a binary tree via a linkage method and then prunes it to enforce validity. We establish necessary and sufficient conditions on the linkage function under which this procedure exactly recovers the finest valid hierarchy, and we show that all linkage functions satisfying these conditions yield the same hierarchy after pruning. Notably, classical linkage rules such as single, complete, and average satisfy these conditions, whereas Ward's linkage fails to do so.

Linkages are essentially determined by similarity measures that may be derived from multiple perspectives. For example, spatial linkages are usually generated based on localities of heterogeneous data, whereas semantic linkages can come from various properties, such as different physical meanings behind social relations. Many existing metric learning models focus on spatial linkages, but leave the rich semantic factors unconsidered. Similarities based on these models are usually overdetermined on linkages. We propose a Unified Multi-Metric Learning (UM2L) framework to exploit multiple types of metrics. In UM2L, a type of combination operator is introduced for distance characterization from multiple perspectives, and thus can introduce flexibilities for representing and utilizing both spatial and semantic linkages. Besides, we propose a uniform solver for UM2L which is guaranteed to converge.

Linkages are essentially determined by similarity measures that may be derived from multiple perspectives. For example, spatial linkages are usually generated based on localities of heterogeneous data, whereas semantic linkages can come from various properties, such as different physical meanings behind social relations. Many existing metric learning models focus on spatial linkages, but leave the rich semantic factors unconsidered. Similarities based on these models are usually overdetermined on linkages.

The main focus of this paper is on minimum spanning tree (MST) based clusterings.

Search and rescue (SAR) robots are required to quickly traverse terrain and perform high-force rescue tasks, necessitating both terrain adaptability and controlled high-force output. Few platforms exist today for SAR, and fewer still have the ability to cover both tasks of terrain adaptability and high-force output when performing extraction. While legged robots offer significant ability to traverse uneven terrain, they typically are unable to incorporate mechanisms that provide variable high-force outputs, unlike traditional wheel-based drive trains. This work introduces a novel concept for a dynamically extensible and retractable robot leg. Leveraging a dynamically extensible and retractable five-bar linkage design, it allows for mechanically switching between height-advantaged and force-advantaged configurations via a geometric transformation. A testbed evaluated leg performance across linkage geometries and operating modes, with empirical and analytical analyses conducted on stride length, force output, and stability. The results demonstrate that the morphing leg offers a promising path toward SAR robots that can both navigate terrain quickly and perform rescue tasks effectively.

The definition of clustering as the task of grouping similar objects emphasizes the importance of assessing similarity scores for the process of clustering.