Ptimization Algorithms On Matrix Manifolds
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Optimization Algorithms on Matrix Manifolds
Author: P.-A. Absil
language: en
Publisher: Princeton University Press
Release Date: 2009-04-11
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. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.
Data Science in Engineering Vol. 10
Data Science in Engineering, Volume 10: Proceedings of the 42nd IMAC, A Conference and Exposition on Structural Dynamics, 2024, the tenth volume of ten from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Data Science in Engineering, including papers on: Novel Data-driven Analysis Methods Deep Learning Gaussian Process Analysis Real-time Video-based Analysis Applications to Nonlinear Dynamics and Damage Detection Data-driven System Prognostics.
Principal Manifolds for Data Visualization and Dimension Reduction
Author: Alexander N. Gorban
language: en
Publisher: Springer Science & Business Media
Release Date: 2007-10
The book starts with the quote of the classical Pearson definition of PCA and includes reviews of various methods: NLPCA, ICA, MDS, embedding and clustering algorithms, principal manifolds and SOM. New approaches to NLPCA, principal manifolds, branching principal components and topology preserving mappings are described. Presentation of algorithms is supplemented by case studies. The volume ends with a tutorial PCA deciphers genome.