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Wireless sensor networks play a pivotal role in a myriad of applications, including agriculture, health monitoring, tracking and structural health monitoring. One crucial aspect of these applications involves accurately determining the positions of the sensors. In this paper, we study a novel Nystrom based sampling protocol in which a selected group of anchor nodes, with known locations, establish communication with only a subset of the remaining sensor nodes. Leveraging partial distance information, we present an efficient algorithm for estimating sensor locations. To demonstrate the effectiveness of our approach, we provide empirical results using synthetic da 

High-dimensional data is commonly encountered in various applications, including genomics, as well as image and video processing. Analyzing, computing, and visualizing such data pose significant challenges. Feature extraction methods are crucial in addressing these challenges by obtaining compressed representations that are suitable for analysis and downstream tasks. One effective technique along these lines is sparse coding, which involves representing data as a sparse linear combination of a set of exemplars. In this study, we propose a local sparse coding framework within the context of a classification problem. The objective is to predict the label of a given data point based on labeled training data. The primary optimization problem encourages the representation of each data point using nearby exemplars. We leverage the optimized sparse representation coefficients to predict the label of a test data point by assessing its similarity to the sparse representations of the training data. The proposed framework is computationally efficient and provides interpretable sparse representations. To illustrate the practicality of our proposed framework, we apply it to agriculture for the classification of crop diseases. 

Classical multidimensional scaling (CMDS) is a technique that embeds a set of objects in a Euclidean space given their pairwise Euclidean distances. The main part of CMDS involves double centering a squared distance matrix and using a truncated eigendecomposition to recover the point coordinates. In this paper, motivated by a study in Euclidean distance geometry, we explore a dual basis approach to CMDS. We give an explicit formula for the dual basis vectors and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness. 

The Euclidean distance geometry (EDG) problem is a crucial machine learning task that appears in many applications. Utilizing the pairwise Euclidean distance information of a given point set, EDG reconstructs the configuration of the point system. When only partial distance information is available, matrix completion techniques can be incorporated to fill in the missing pairwise distances. In this paper, we propose a novel dual basis Riemannian gradient descent algorithm, coined RieEDG, for the EDG completion problem. The numerical experiments verify the effectiveness of the proposed algorithm. In particular, we show that RieEDG can precisely reconstruct various datasets consisting of 2- and 3-dimensional points by accessing a small fraction of pairwise distance information. 

A bstract The identification of interesting substructures within jets is an important tool for searching for new physics and probing the Standard Model at colliders. Many of these substructure tools have previously been shown to take the form of optimal transport problems, in particular the Energy Mover’s Distance (EMD). In this work, we show that the EMD is in fact the natural structure for comparing collider events, which accounts for its recent success in understanding event and jet substructure. We then present a Shape Hunting Algorithm using Parameterized Energy Reconstruction (S haper ), which is a general framework for defining and computing shape-based observables. S haper generalizes N -jettiness from point clusters to any extended, parametrizable shape. This is accomplished by efficiently minimizing the EMD between events and parameterized manifolds of energy flows representing idealized shapes, implemented using the dual-potential Sinkhorn approximation of the Wasserstein metric. We show how the geometric language of observables as manifolds can be used to define novel observables with built-in infrared-and-collinear safety. We demonstrate the efficacy of the S haper framework by performing empirical jet substructure studies using several examples of new shape-based observables.