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 can not 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 ν, the main result shows that the underlying configuration of n points can be recovered with very high probability from O(nrν log 2 (n)) uniformly random samples. Computationally, simple and fast algorithms are designed to solve the Euclidean distance geometry problem. Numerical tests on different three dimensional data and protein molecules validate effectiveness and efficiency of the proposed algorithms. Index TermsEuclidean distance geometry, low-rank matrix completion, nuclear norm minimization, dual basis, random matrices, Gram matrix.X · 1 = 0 ; X = X T ; X 0 A. Tasissa is with the 2 Here ||X|| * denotes the nuclear norm and Ω ⊂ {(i, j)|i, j = 1, ..., n, i < j}, |Ω| = m, denotes the random set that consists of all the sampled indices. One characterization of the Euclidean distance matrix, due to Gower [9], states that D is an Euclidean distance matrix if and only if D = X i ,i +X j , j −2X i , j = D i , j ∀i, j for some positive semidefinite matrix X satisfying X ·1 = 0. As such, the above nuclear norm minimization can be interpreted as a regularization of the rank with a prior assumption that the true Gram matrix is low rank. An alternative approach based on the matrix completion method is direct completion of the distance matrix [10], [11], [12]. Compared to this approach, an advantage of the above minimization program can be seen by comparing the rank of the Gram matrix X and the distance matrix D. Using (1), the rank of D is at most r + 2 while the rank of X is simply r. Using matrix completion theory, it can be surmised that relatively less number of samples are required for the Gram matrix completion. Numerical experiments in [7] confirm this observation. In [13], the authors consider a theoretical analysis of a specific instance of localization problem and propose an algorithm similar to (2). The paper considers a random geometric model and derives interesting results of bound of errors in reconstructing point coordinates. While the localization problem and EDG problem share a common theme, we remark that the EDG problem is different and our analysis adopts the matrix completion framework. The main task of this paper is a theoretical analysis of the above minimization problem. In particular, under appropriate conditions, we will show that the above nuclear norm minimization recovers the underlying inner product mat...
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 semianalytical 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.
A method for active learning of hyperspectral images (HSI) is proposed, which combines deep learning with diffusion processes on graphs. A deep variational autoencoder extracts smoothed, denoised features from a high-dimensional HSI, which are then used to make labeling queries based on graph diffusion processes. The proposed method combines the robust representations of deep learning with the mathematical tractability of diffusion geometry, and leads to strong performance on real HSI.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.