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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. [ABSTRACT FROM AUTHOR]

This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with sampling under entry sensing or with respect to a symmetric orthonormal basis. The analysis is based on dual certificates using a dual basis approach. We introduce a condition on the basis called the correlation condition. This condition can be computed in time O (n 3) and holds for many cases of deterministic basis. If the correlation condition holds and the underlying low rank matrix obeys the coherence condition with parameter ν , under additional mild assumptions, our main result shows that the true matrix can be recovered with very high probability from O (n r ν log 2 ⁡ n) uniformly random expansion coefficients. [ABSTRACT FROM AUTHOR]

The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. When the distance information is incomplete, the problem can be formulated as a nuclear norm minimization problem. In this paper, this minimization program is recast as a matrix completion problem of a low-rank $r$ Gram matrix with respect to a suitable basis. The well-known restricted isometry property cannot be satisfied in this scenario. Instead, a dual basis approach is introduced to theoretically analyze the reconstruction problem. If the Gram matrix satisfies certain coherence conditions with parameter $\nu $ , the main result shows that the underlying configuration of $n$ points can be recovered with very high probability from $O(nr\nu \log ^{2}(n))$ uniformly random samples. Computationally, simple and fast algorithms are designed to solve the Euclidean distance geometry problem. Numerical tests on different 3-D data and protein molecules validate the effectiveness and efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Compressed-gas-driven shock tubes have become popular as a laboratory-scale replacement for field blast tests. The well-known initial structure of the Riemann problem eventually evolves into a shock structure thought to resemble a Friedlander wave, although this remains to be demonstrated theoretically. In this paper, we develop a semi-analytical model to predict the key characteristics of pseudo blast waves forming in a shock tube: location where the wave first forms, peak overpressure, decay time and impulse. The approach is based on combining the solutions of the two different types of wave interactions that arise in the shock tube after the family of rarefaction waves in the Riemann solution interacts with the closed end of the tube. The results of the analytical model are verified against numerical simulations obtained with a finite volume method. The model furnishes a rational approach to relate shock tube parameters to desired blast wave characteristics, and thus constitutes a useful tool for the design of shock tubes for blast testing. [ABSTRACT FROM AUTHOR]

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 (Shaper), which is a general framework for defining and computing shape-based observables. Shaper 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 Shaper framework by performing empirical jet substructure studies using several examples of new shape-based observables. [ABSTRACT FROM AUTHOR]