Note havethatd(z, w)=( y, z)w ifandonlyd(z, w)=( x, z)w. InProceedingsof the Twenty-sixth Annual ACMSymposiumon Principlesof Distributed Computing, PODC '07, pages 43-52, New York, NY, USA, 2007.

Sliding windows and persistence: an application of topological methods to signal analysis.

However, in many applications there are several different parameters one might wish to vary: for example, scale and density.

Abstract-- A common robotics sensing problem is to place sensors to robustly monitor a set of assets, where robustness is assured by requiring asset p to be monitored by at least κ (p) sensors. Given n assets that must be observed by m sensors, each with a disk-shaped sensing region, where should the sensors be placed to minimize the total area observed? W e provide and analyze a fast heuristic for this problem. Subsequently, we enforce separation constraints between the sensors by modifying the integer program formulation and by changing the disk candidate set. I. Introduction Coordinating different kinds of robotic assets is a natural challenge when it comes to problems of surveillance and guidance. As shown in Figure 1, this gives rise to scenarios in which a finite set of drones with downward communication links must maintain control of a finite set of ground assets [7], [15], choosing a set of different altitudes that balance safe separation between drones with reliable communication to the ground. The latter requires sufficient signal strength, so communication areas (and thus energy consumption) depend quadratically on the altitude.

Binary classification is a fundamental task in machine learning and data science, with applications spanning numerous domains, including spam detection, medical diagnostics, image recognition, credit scoring. The goal is to predict a binary outcome--typically labeled as 0 or 1--based on a set of input features. Various machine learning algorithms, such as logistic regression (LR), k-nearest neighbors (kNN), support vector machines (SVM), and neural network (NN), are commonly employed for binary classification tasks. These algorithms can be mainly divided into two categories: those with interpretability, which are convenient for analysis and control (e.g., LR); and those without interpretability but with potentially good classification performance (e.g., NN). Ensemble learning, a powerful technique in predictive modeling, has gained widespread recognition for its ability to improve model performance by combining the strengths of multiple learning algorithms [1]. Among ensemble methods, stacking stands out by integrating the predictions of diverse base models (different learning algorithms) through a meta-model, resulting in enhanced prediction accuracy compared to only using the best base model [2]. Stacking has demonstrated significant applications in classification problems such as network intrusion detection [3, 4], cancer type classification [5], credit lending [6], and protein-protein binding affinity prediction [7].

VC-dimension and $\varepsilon$-nets are key concepts in Statistical Learning Theory. Intuitively, VC-dimension is a measure of the size of a class of sets. The famous $\varepsilon$-net theorem, a fundamental result in Discrete Geometry, asserts that if the VC-dimension of a set system is bounded, then a small sample exists that intersects all sufficiently large sets. In online learning scenarios where data arrives sequentially, the VC-dimension helps to bound the complexity of the set system, and $\varepsilon$-nets ensure the selection of a small representative set. This sampling framework is crucial in various domains, including spatial data analysis, motion planning in dynamic environments, optimization of sensor networks, and feature extraction in computer vision, among others. Motivated by these applications, we study the online $\varepsilon$-net problem for geometric concepts with bounded VC-dimension. While the offline version of this problem has been extensively studied, surprisingly, there are no known theoretical results for the online version to date. We present the first deterministic online algorithm with an optimal competitive ratio for intervals in $\mathbb{R}$. Next, we give a randomized online algorithm with a near-optimal competitive ratio for axis-aligned boxes in $\mathbb{R}^d$, for $d\le 3$. Furthermore, we introduce a novel technique to analyze similar-sized objects of constant description complexity in $\mathbb{R}^d$, which may be of independent interest. Next, we focus on the continuous version of this problem, where ranges of the set system are geometric concepts in $\mathbb{R}^d$ arriving in an online manner, but the universe is the entire space, and the objective is to choose a small sample that intersects all the ranges.

The problem of modeling ropes arises in many applications, including providing haptic feedback to surgeons who are using surgical robots to realign the distal and proximal ends of split bones. Here, we consider a simplified, 2D variant of the haptic feedback estimation problem and discuss how visibility decompositions greatly simplify the problem. Then, we introduce an efficient, concise algorithm for modeling the dynamics of 2D ropes around polygonal obstacles in O(n) time, where n is the number of line segment obstacles. We start by providing a brief definition of our 2D rope problem. The open line segment obstacles constitute C entirely.

How can we train a statistical mixture model on a massive data set? In this paper, we show how to construct coresets for mixtures of Gaussians and natural generalizations. A coreset is a weighted subset of the data, which guarantees that models fitting the coreset will also provide a good fit for the original data set. We show that, perhaps surprisingly, Gaussian mixtures admit coresets of size independent of the size of the data set.