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Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.

Background Screening programs use mammography as primary diagnostic tool for detecting breast cancer at an early stage. The diagnosis of some lesions, such as microcalcifications, is still difficult today for radiologists. In this paper, we proposed an automatic binary model for discriminating tissue in digital mammograms, as support tool for the radiologists. In particular, we compared the contribution of different methods on the feature selection process in terms of the learning performances and selected features. Results For each ROI, we extracted textural features on Haar wavelet decompositions and also interest points and corners detected by using Speeded Up Robust Feature (SURF) and Minimum Eigenvalue Algorithm (MinEigenAlg). Then a Random Forest binary classifier is trained on a subset of a sub-set features selected by two different kinds of feature selection techniques, such as filter and embedded methods. We tested the proposed model on 260 ROIs extracted from digital mammograms of the BCDR public database. The best prediction performance for the normal/abnormal and benign/malignant problems reaches a median AUC value of 98.16 % and 92.08 % , and an accuracy of 97.31 % and 88.46 % , respectively. The experimental result was comparable with related work performance. Conclusions The best performing result obtained with embedded method is more parsimonious than the filter one. The SURF and MinEigen algorithms provide a strong informative content useful for the characterization of microcalcification clusters.

In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued θ-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented.

Ordinary differential equations (ODEs) are widely used to model the dynamic behavior of a complex system. Parameter estimation and variable selection for a \"Big System\" with linear ODEs are very challenging due to the need of nonlinear optimization in an ultra-high dimensional parameter space. In this article, we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares (SLS). Simulation studies demonstrate that the proposed matrix-based SLS method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ODE system with thousands of dimensions and millions of parameters much better than the direct least squares method and the vector-based two-stage method that are currently available. We applied this new method to two real datasets-a yeast cell cycle gene expression dataset with 30 dimensions and 930 unknown parameters and the Standard & Poor 1500 index stock price data with 1250 dimensions and 1,563,750 unknown parameters-to illustrate the utility and numerical performance of the proposed parameter estimation and variable selection method for big systems in practice. Supplementary materials for this article are available online.

In this paper, a new application of the Numerical Assembly Technique is presented for the balancing of linear elastic rotor-bearing systems with a stepped shaft and arbitrarily distributed mass unbalance. The method improves existing balancing techniques by combining the advantages of modal balancing with the fast calculation of an efficient numerical method. The rotating stepped circular shaft is modelled according to the Rayleigh beam theory. The Numerical Assembly Technique is used to calculate the steady-state harmonic response, eigenvalues and the associated mode shapes of the rotor. The displacements of a simulation are compared to measured displacements of the rotor-bearing system to calculate the generalized unbalance for each eigenvalue. The generalized unbalances are modified according to modal theory to calculate orthogonal correction masses. In this manner, a rotor-bearing system is balanced using a single measurement of the displacement at one position on the rotor for every critical speed. Three numerical examples are used to show the accuracy and the balancing success of the proposed method.

This study examines the optimization of digital resource management and sharing in preschool education, employing the Goodness of Fit (GoF) algorithm to assess eigenvalues’ suitability in this context. As digital educational materials proliferate, effectively managing and disseminating these resources is a critical challenge. We have developed a preschool educational digital resource management and sharing system grounded in GoF algorithmic principles through an in-depth evaluation of various GoF algorithms’ classification and detection capabilities. Our methodology combines theoretical exploration with empirical algorithm performance testing. Findings indicate that a GoF algorithm, explicitly leveraging the chi-square distribution, achieves over 95% accuracy in identifying preschool digital resources. Furthermore, this study introduces a novel dynamic fitting object model, addressing the GoF algorithm’s adaptability in fluctuating environments. Results affirm the algorithm’s broad adaptability and promising efficiency for managing and sharing digital resources in preschool education settings.

This book lays out the theoretical groundwork for personalized search and reputation management, both on the Web and in peer-to-peer and social networks. Representing much of the foundational research in this field, the book develops scalable algorithms that exploit the graphlike properties underlying personalized search and reputation management, and delves into realistic scenarios regarding Web-scale data. Sep Kamvar focuses on eigenvector-based techniques in Web search, introducing a personalized variant of Google's PageRank algorithm, and he outlines algorithms--such as the now-famous quadratic extrapolation technique--that speed up computation, making personalized PageRank feasible. Kamvar suggests that Power Method-related techniques ultimately should be the basis for improving the PageRank algorithm, and he presents algorithms that exploit the convergence behavior of individual components of the PageRank vector. Kamvar then extends the ideas of reputation management and personalized search to distributed networks like peer-to-peer and social networks. He highlights locality and computational considerations related to the structure of the network, and considers such unique issues as malicious peers. He describes the EigenTrust algorithm and applies various PageRank concepts to P2P settings. Discussion chapters summarizing results conclude the book's two main sections. Clear and thorough, this book provides an authoritative look at central innovations in search for all of those interested in the subject.