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We propose an algorithm to find an integer point of a simplex. The algorithm is based on an integer labeling rule and a triangulation of the space. Starting at an integer point, the algorithm leaves it along one of n rays and generates a sequence of adjacent simplices of varying dimension. Within a finite number of iterations, the algorithm either yields an integer point of the simplex or proves that no such point exists.

An important problem in constrained optimization is to determine whether or not a vector can be represented as the conical combination (i.e., linear nonnegative combination) of some given vectors. This problem can be transformed into a special linear programming problem (SLP). A new approach, the variable-dimension boundary-point algorithm (VDBPA), based on the projection of a vector into a variable-dimension subspace, is proposed to solve this problem. When a conical combination representation (CCR) of a vector exists, the VDBPA can compute its CCR coefficients; otherwise, the algorithm certificates the nonexistence of such representation. In order to assure convergence, the VDBPA may call the lexicographically ordered method (LOM) for linear programming at the final stage. In fact, the VDBPA terminates often by solving SLP for most instances before calling the LOM. Numerical results indicate that the VDBPA works more efficiently than the LOM for problems that have more variables or inequality constraints. Also, we have found instances of the SLP, when the number of inequality constraints is about twice the number of variables, which are much more difficult to solve than other instances. In addition, the convergence of the VDBPA without calling the LOM is established under certain conditions.

We consider here the problem of solving a system of n nonlinear equations in n variables, when n is large, but the underlying mapping has a sparse Jacobian, and is also structured. We present a homotopy, having a variable dimension feature, whose implementation in a PL algorithm effectively exploits the sparsity of the Jacobian and separability of the mapping. The implementation given here uses the Cholesky factorization and is thus stable. An application to a large system is also discussed.

This accessible book provides an introduction to the analysis and design of dynamic multiagent networks. Such networks are of great interest in a wide range of areas in science and engineering, including: mobile sensor networks, distributed robotics such as formation flying and swarming, quantum networks, networked economics, biological synchronization, and social networks. Focusing on graph theoretic methods for the analysis and synthesis of dynamic multiagent networks, the book presents a powerful new formalism and set of tools for networked systems. The book's three sections look at foundations, multiagent networks, and networks as systems. The authors give an overview of important ideas from graph theory, followed by a detailed account of the agreement protocol and its various extensions, including the behavior of the protocol over undirected, directed, switching, and random networks. They cover topics such as formation control, coverage, distributed estimation, social networks, and games over networks. And they explore intriguing aspects of viewing networks as systems, by making these networks amenable to control-theoretic analysis and automatic synthesis, by monitoring their dynamic evolution, and by examining higher-order interaction models in terms of simplicial complexes and their applications. The book will interest graduate students working in systems and control, as well as in computer science and robotics. It will be a standard reference for researchers seeking a self-contained account of system-theoretic aspects of multiagent networks and their wide-ranging applications. This book has been adopted as a textbook at the following universities: University of Stuttgart, GermanyRoyal Institute of Technology, SwedenJohannes Kepler University, AustriaGeorgia Tech, USAUniversity of Washington, USAOhio University, USA

Change points are a very common feature of 'big data' that arrive in the form of a data stream. We study high dimensional time series in which, at certain time points, the mean structure changes in a sparse subset of the co-ordinates. The challenge is to borrow strength across the co-ordinates to detect smaller changes than could be observed in any individual component series. We propose a two-stage procedure called inspect for estimation of the change points: first, we argue that a good projection direction can be obtained as the leading left singular vector of the matrix that solves a convex optimization problem derived from the cumulative sum transformation of the time series. We then apply an existing univariate change point estimation algorithm to the projected series. Our theory provides strong guarantees on both the number of estimated change points and the rates of convergence of their locations, and our numerical studies validate its highly competitive empirical performance for a wide range of data-generating mechanisms. Software implementing the methodology is available in the R package InspectChangepoint.

The authors introduce a new low-distortion embedding of ... into ... called the fast Johnson-Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. They overcome this handicap by exploiting the \"Heisenberg principle\" of the Fourier transform, i.e., its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in ... and ... . The consider the case of approximate nearest neighbors in ... . They provide a faster algorithm using classical projections, which they then speed up further by plugging in the FJLT.They also give a faster algorithm for searching over the hypercube. (ProQuest: ... denotes formulae/symbols omitted.)

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

With the progress of measurement apparatus and the development of automatic sensors, it is not unusual anymore to get large samples of observations taking values in high-dimension spaces, such as functional spaces. In such large samples of high-dimensional data, outlying curves may not be uncommon, and even a few individuals may corrupt simple statistical indicators, such as the mean trajectory. We focus here on the estimation of the geometric median which is a direct generalization of the real median in metric spaces and has nice robustness properties. It is possible to estimate the geometric median, being defined as the minimizer of a simple convex functional that is differentiable everywhere when the distribution has no atom, with online gradient algorithms. Such algorithms are very fast and can deal with large samples. Furthermore, they also can be simply updated when the data arrive sequentially. We state the almost sure consistency and the L 2 rates of convergence of the stochastic gradient estimator as well as the asymptotic normality of its averaged version. We get that the asymptotic distribution of the averaged version of the algorithm is the same as the classic estimators, which are based on the minimization of the empirical loss function. The performances of our averaged sequential estimator, both in terms of computation speed and accuracy of the estimations, are evaluated with a small simulation study. Our approach is also illustrated on a sample of more than 5000 individual television audiences measured every second over a period of 24 hours.

By combining the thin QR decomposition and the subsampled randomized Fourier transform (SRFT), we obtain an efficient randomized algorithm for computing the approximate Tucker decomposition with a given target multilinear rank. We also combine this randomized algorithm with the power iteration technique to improve the efficiency of the algorithm. By using the results about the singular values of the product of orthonormal matrices with the Kronecker product of SRFT matrices, we obtain the error bounds of these two algorithms. Finally, the efficiency of these algorithms is illustrated by several numerical examples.

Background Rapid computational and technological developments made large amounts of omics data available in different biological levels. It is becoming clear that simultaneous data analysis methods are needed for better interpretation and understanding of the underlying systems biology. Different methods have been proposed for this task, among them Partial Least Squares (PLS) related methods. To also deal with orthogonal variation, systematic variation in the data unrelated to one another, we consider the Two-way Orthogonal PLS (O2PLS): an integrative data analysis method which is capable of modeling systematic variation, while providing more parsimonious models aiding interpretation. Results A simulation study to assess the performance of O2PLS showed positive results in both low and higher dimensions. More noise (50 % of the data) only affected the systematic part estimates. A data analysis was conducted using data on metabolomics and transcriptomics from a large Finnish cohort (DILGOM). A previous sequential study, using the same data, showed significant correlations between the Lipo-Leukocyte (LL) module and lipoprotein metabolites. The O2PLS results were in agreement with these findings, identifying almost the same set of co-varying variables. Moreover, our integrative approach identified other associative genes and metabolites, while taking into account systematic variation in the data. Including orthogonal components enhanced overall fit, but the orthogonal variation was difficult to interpret. Conclusions Simulations showed that the O2PLS estimates were close to the true parameters in both low and higher dimensions. In the presence of more noise (50 %), the orthogonal part estimates could not distinguish well between joint and unique variation. The joint estimates were not systematically affected. Simultaneous analysis with O2PLS on metabolome and transcriptome data showed that the LL module, together with VLDL and HDL metabolites, were important for the metabolomic and transcriptomic relation. This is in agreement with an earlier study. In addition more gene expression and metabolites are identified being important for the joint covariation.