Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

A D.C. Programming Approach to the Sparse Generalized Eigenvalue Problem

In this paper, we consider the sparse eigenvalue problem wherein the goal is to obtain a sparse solution to the generalized eigenvalue problem. We achieve this by constraining the cardinality of the solution to the generalized eigenvalue problem and obtain sparse principal component analysis (PCA), sparse canonical correlation analysis (CCA) and sparse Fisher discriminant analysis (FDA) as special cases. Unlike the $\ell_1$-norm approximation to the cardinality constraint, which previous methods have used in the context of sparse PCA, we propose a tighter approximation that is related to the negative log-likelihood of a Student's t-distribution. The problem is then framed as a d.c. (difference of convex functions) program and is solved as a sequence of convex programs by invoking the majorization-minimization method. The resulting algorithm is proved to exhibit \emph{global convergence} behavior, i.e., for any random initialization, the sequence (subsequence) of iterates generated by the algorithm converges to a stationary point of the d.c. program. The performance of the algorithm is empirically demonstrated on both sparse PCA (finding few relevant genes that explain as much variance as possible in a high-dimensional gene dataset) and sparse CCA (cross-language document retrieval and vocabulary selection for music retrieval) applications.Comment: 40 page

Rank-Two Beamforming and Power Allocation in Multicasting Relay Networks

In this paper, we propose a novel single-group multicasting relay beamforming scheme. We assume a source that transmits common messages via multiple amplify-and-forward relays to multiple destinations. To increase the number of degrees of freedom in the beamforming design, the relays process two received signals jointly and transmit the Alamouti space-time block code over two different beams. Furthermore, in contrast to the existing relay multicasting scheme of the literature, we take into account the direct links from the source to the destinations. We aim to maximize the lowest received quality-of-service by choosing the proper relay weights and the ideal distribution of the power resources in the network. To solve the corresponding optimization problem, we propose an iterative algorithm which solves sequences of convex approximations of the original non-convex optimization problem. Simulation results demonstrate significant performance improvements of the proposed methods as compared with the existing relay multicasting scheme of the literature and an algorithm based on the popular semidefinite relaxation technique

Maxent-Stress Optimization of 3D Biomolecular Models

Knowing a biomolecule\u27s structure is inherently linked to and a prerequisite for any detailed understanding of its function. Significant effort has gone into developing technologies for structural characterization. These technologies do not directly provide 3D structures; instead they typically yield noisy and erroneous distance information between specific entities such as atoms or residues, which have to be translated into consistent 3D models. Here we present an approach for this translation process based on maxent-stress optimization. Our new approach extends the original graph drawing method for the new application\u27s specifics by introducing additional constraints and confidence values as well as algorithmic components. Extensive experiments demonstrate that our approach infers structural models (i.e., sensible 3D coordinates for the molecule\u27s atoms) that correspond well to the distance information, can handle noisy and error-prone data, and is considerably faster than established tools. Our results promise to allow domain scientists nearly-interactive structural modeling based on distance constraints

Landscape genetic connectivity in a riparian foundation tree is jointly driven by climatic gradients and river networks

Fremont cottonwood (Populus fremonti) is a foundation riparian tree species that drives community structure and ecosystem processes in southwestern U.S. ecosystems. Despite its ecological importance, little is known about the ecological and environmental processes that shape its genetic diversity, structure, and landscape connectivity. Here, we combined molecular analyses of 82 populations including 1312 individual trees dispersed over the species’ geographical distribution. We reduced the data set to 40 populations and 743 individuals to eliminate admixture with a sibling species, and used multivariate restricted optimization and reciprocal causal modeling to evaluate the effects of river network connectivity and climatic gradients on gene flow. Our results confirmed the following: First, gene flow of Fremont cottonwood is jointly controlled by the connectivity of the river network and gradients of seasonal precipitation. Second, gene flow is facilitated by mid-sized to large rivers, and is resisted by small streams and terrestrial uplands, with resistance to gene flow decreasing with river size. Third, genetic differentiation increases with cumulative differences in winter and spring precipitation. Our results suggest that ongoing fragmentation of riparian habitats will lead to a loss of landscape-level genetic connectivity, leading to increased inbreeding and the concomitant loss of genetic diversity in a foundation species. These genetic effects will cascade to a much larger community of organisms, some of which are threatened and endangered