Structure Preserving Dimension Reduction Research Articles

Two-sided dimension reduction for linear control systems of [formula omitted]th-order form

In this article, a two-sided structure-preserving dimension reduction method for linear control systems of kth-order form is discussed. We first recall the Petrov–Galerkin framework of dimension reduction for the kth linear systems. Then the moments of systems at a given frequency point s0 are discussed. According to the recurrence form of the moments, we obtain the input Krylov subspace and the projection matrix V. Via dually designing output Krylov subspace, we obtain the output Krylov subspace basis matrix W. Hence, the two-sided dimension reduction method is obtained via combination V with W. Afterwards one can obtain the result that the number of moments matched is twice as much as the column dimension of the projection matrix. Finally, one numerical experiment is performed to verify the proposed dimension reduction method.

Discriminative geodesic Gaussian process latent variable model for structure preserving dimension reduction in clustering and classification problems

Dimension reduction is a common approach for analyzing complex high-dimensional data and allows efficient implementation of classification and decision algorithms. Gaussian process latent variable model (GPLVM) is a widely applicable dimension reduction method which represents latent space without considering the class labels. Preserving the structure and topology of data are key factors that influence the performance of dimensionality reduction models. A conventional measure which reflects the topological structure of data points is geodesic distance. In this study, we propose an enriched GPLVM mapping between low-dimensional space and high-dimensional data. One of the contributions of the proposed approach is to calculate geodesic distance under the influence of class labels and introducing an improved GPLVM kernel using the distance. Also, the objective function of the model is reformulated to consider the trade-off between class separation and structure preservation which improves discrimination power and compactness of data. The efficiency of the proposed approach is compared with other dimension reduction techniques such as the kernel principal component analysis (KPCA), locally linear embedding (LLE), Laplacian eigenmaps and also discriminative and supervised extensions of standard GPLVM. Based on the experiments, it is suggested that the proposed model has a higher capacity for accurate classification and clustering of data as compared with the mentioned approaches.

Shape component analysis: structure-preserving dimension reduction on biological shape spaces.

Quantitative shape analysis is required by a wide range of biological studies across diverse scales, ranging from molecules to cells and organisms. In particular, high-throughput and systems-level studies of biological structures and functions have started to produce large volumes of complex high-dimensional shape data. Analysis and understanding of high-dimensional biological shape data require dimension-reduction techniques. We have developed a technique for non-linear dimension reduction of 2D and 3D biological shape representations on their Riemannian spaces. A key feature of this technique is that it preserves distances between different shapes in an embedded low-dimensional shape space. We demonstrate an application of this technique by combining it with non-linear mean-shift clustering on the Riemannian spaces for unsupervised clustering of shapes of cellular organelles and proteins. Source code and data for reproducing results of this article are freely available at https://github.com/ccdlcmu/shape_component_analysis_Matlab The implementation was made in MATLAB and supported on MS Windows, Linux and Mac OS. geyang@andrew.cmu.edu.

Sparse-Representation-Based Classification with Structure-Preserving Dimension Reduction

Sparse-representation-based classification (SRC), which classifies data based on the sparse reconstruction error, has been a new technique in pattern recognition. However, the computation cost for sparse coding is heavy in real applications. In this paper, various dimension reduction methods are studied in the context of SRC to improve classification accuracy as well as reduce computational cost. A feature extraction method, i.e., principal component analysis, and feature selection methods, i.e., Laplacian score and Pearson correlation coefficient, are applied to the data preparation step to preserve the structure of data in the lower-dimensional space. Classification performance of SRC with structure-preserving dimension reduction (SRC–SPDR) is compared to classical classifiers such as k-nearest neighbors and support vector machines. Experimental tests with the UCI and face data sets demonstrate that SRC–SPDR is effective with relatively low computation cost

Dimension reduction for second-order systems by general orthogonal polynomials

In this article, we discuss the time-domain dimension reduction methods for second-order systems by general orthogonal polynomials, and present a structure-preserving dimension reduction method for second-order systems. The resulting reduced systems not only preserve the second-order structure but also guarantee the stability under certain conditions. The error estimate of the reduced models is also given. The effectiveness of the proposed methods is demonstrated by three test examples.

Dimension Reduction of Large-Scale Second-Order Dynamical Systems via a Second-Order Arnoldi Method

A structure-preserving dimension reduction algorithm for large-scale second-order dynamical systems is presented. It is a projection method based on a second-order Krylov subspace. A second-order Arnoldi (SOAR) method is used to generate an orthonormal basis of the projection subspace. The reduced system not only preserves the second-order structure but also has the same order of approximation as the standard Arnoldi-based Krylov subspace method via linearization. The superior numerical properties of the SOAR-based method are demonstrated by examples from structural dynamics and microelectromechanical systems.