Clustering algorithms remain valuable tools for grouping and summarizing the most important aspects of data. Example areas where this is the case include image segmentation, dimension reduction, signals analysis, model order reduction, numerical analysis, and others. As a consequence, many clustering approaches have been developed to satisfy the unique needs of each particular field. In this article, we present a family of data-adaptive partitioning algorithms that unifies several well-known methods (e.g., k-means and k-subspaces). Indexed by a single parameter and employing a common minimization strategy, the algorithms are easy to use and interpret, and scale well to large, high-dimensional problems. In addition, we develop an adaptive mechanism that (a) exhibits skill at automatically uncovering data structures and problem parameters without any expert knowledge and, (b) can be used to augment other existing methods. By demonstrating the performance of our methods on examples from disparate fields including subspace clustering, model order reduction, and matrix approximation, we hope to highlight their versatility and potential for extending the boundaries of existing scientific domains. We believe our family's parametrized structure represents a synergism of algorithms that will foster new developments and directions, not least within the data science community.

Multi-robot coverage is crucial in numerous applications, including environmental monitoring, search and rescue operations, and precision agriculture. In modern applications, a multi-robot team must collaboratively explore unknown spatial fields in GPS-denied and extreme environments where global localization is unavailable. Coverage algorithms typically assume that the robot positions and the coverage environment are defined in a global reference frame. However, coordinating robot motion and ensuring coverage of the shared convex workspace without global localization is challenging. This paper proposes a novel anchor-oriented coverage (AOC) approach to generate dynamic localized Voronoi partitions based around a common anchor position. We further propose a consensus-based coordination algorithm that achieves agreement on the coverage workspace around the anchor in the robots' relative frames of reference. Through extensive simulations and real-world experiments, we demonstrate that the proposed anchor-oriented approach using localized Voronoi partitioning performs as well as the state-of-the-art coverage controller using GPS.

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions, with BNQN. In general, they look more smooth than that of Newton's method. In this paper, we continue to experimentally explore in depth this remarkable phenomenon, and connect BNQN to Newton's flow and Voronoi's diagram. This link poses a couple of challenging puzzles to be explained. Experiments also indicate that BNQN is more robust against random perturbations than Newton's method and Random Relaxed Newton's method.

Gradient-based sampling algorithms have demonstrated their effectiveness in text generation, especially in the context of controlled text generation. However, there exists a lack of theoretically grounded and principled approaches for this task. In this paper, we take an important step toward building a principled approach for sampling from language models with gradient-based methods. We use discrete distributions given by language models to define densities and develop an algorithm based on Hamiltonian Monte Carlo to sample from them. We name our gradient-based technique Structured Voronoi Sampling (SVS). In an experimental setup where the reference distribution is known, we show that the empirical distribution of SVS samples is closer to the reference distribution compared to alternative sampling schemes. Furthermore, in a controlled generation task, SVS is able to generate fluent and diverse samples while following the control targets significantly better than other methods.

This paper examines the spatial coverage optimization problem for multiple sensors in a known convex environment, where the coverage service of each sensor is heterogeneous and anisotropic. We introduce the Stein Coverage algorithm, a distribution-matching coverage approach that aims to place sensors at positions and orientations such that their collective coverage distribution is as close as possible to the event distribution. To select the most important representative points from the coverage event distribution, Stein Coverage utilizes the Stein Variational Gradient Descent (SVGD), a deterministic sampling method from the variational inference literature. An innovation in our work is the introduction of a repulsive force between the samples in the SVGD algorithm to spread the samples and avoid footprint overlap for the deployed sensors. After pinpointing the points of interest for deployment, Stein Coverage solves the multisensor assignment problem using a bipartite optimal matching process. Simulations demonstrate the advantages of the Stein Coverage method compared to conventional Voronoi partitioning multisensor deployment methods.

In this paper, we present algorithms to identify environmental hotspots using mobile sensors. We examine two approaches: one involving a single robot and another using multiple robots coordinated through a decentralized robot system. We introduce an adaptive algorithm that does not require precise knowledge of Gaussian Processes (GPs) hyperparameters, making the modeling process more flexible. The robots operate for a pre-defined time in the environment. The multi-robot system uses Voronoi partitioning to divide tasks and a Monte Carlo Tree Search for optimal path planning. Our tests on synthetic and a real-world dataset of Chlorophyll density from a Pacific Ocean sub-region suggest that accurate estimation of GP hyperparameters may not be essential for hotspot detection, potentially simplifying environmental monitoring tasks.

Multi-agent patrolling is a key problem in a variety of domains such as intrusion detection, area surveillance, and policing which involves repeated visits by a group of agents to specified points in an environment. While the problem is well-studied, most works either do not consider agent attrition or impose significant communication requirements to enable adaptation. In this work, we present the Adaptive Heuristic-based Patrolling Algorithm, which is capable of adaptation to agent loss using minimal communication by taking advantage of Voronoi partitioning. Additionally, we provide new centralized and distributed mathematical programming formulations of the patrolling problem, analyze the properties of Voronoi partitioning, and show the value of our adaptive heuristic algorithm by comparison with various benchmark algorithms using a realistic simulation environment based on the Robot Operating System (ROS) 2.

Perception of the full state is an essential technology to support the monitoring, analysis, and design of physical systems, one of whose challenges is to recover global field from sparse observations. Well-known for brilliant approximation ability, deep neural networks have been attractive to data-driven flow and heat field reconstruction studies. However, limited by network structure, existing researches mostly learn the reconstruction mapping in finite-dimensional space and has poor transferability to variable resolution of outputs. In this paper, we extend the new paradigm of neural operator and propose an end-to-end physical field reconstruction method with both excellent performance and mesh transferability named RecFNO. The proposed method aims to learn the mapping from sparse observations to flow and heat field in infinite-dimensional space, contributing to a more powerful nonlinear fitting capacity and resolution-invariant characteristic. Firstly, according to different usage scenarios, we develop three types of embeddings to model the sparse observation inputs: MLP, mask, and Voronoi embedding. The MLP embedding is propitious to more sparse input, while the others benefit from spatial information preservation and perform better with the increase of observation data. Then, we adopt stacked Fourier layers to reconstruct physical field in Fourier space that regularizes the overall recovered field by Fourier modes superposition. Benefiting from the operator in infinite-dimensional space, the proposed method obtains remarkable accuracy and better resolution transferability among meshes. The experiments conducted on fluid mechanics and thermology problems show that the proposed method outperforms existing POD-based and CNN-based methods in most cases and has the capacity to achieve zero-shot super-resolution.

We propose a new algorithm for the $k$-means problem which repeatedly increases and decreases the number of centroids by $m$ in order to find an approximate solution. New centroids are inserted in areas where they will likely reduce the error. The subsequent removal of centroids is done such that the resulting raise in error is small. After each increase or decrease step standard $k$-means is performed. Termination is guaranteed by decrementing $m$ after each increase/decrease cycle unless the overall error was lowered. In experiments with Gaussian mixture distributions the new algorithm produced on average solutions several percent better than $k$-means++.