The singular value decomposition and applications in geodesy

The paper considers the singular value decomposition (SVD) of a general matrix. Some immediate applications, such as determining the spectral and Frobenius norm, rank and pseudoinverse of the matrix are described. Applications also include approximating the given matrix by a matrix of a lower rank. It is also shown how to use SVD for solving the homogeneous linear system and the least squares problem. The paper consists of three parts: 1.) The singular value decomposition, 2.) Some applications of the singular value decomposition, 3.) Applications in geodesy

Empirical Evaluation of Four Tensor Decomposition Algorithms

Higher-order tensor decompositions are analogous to the familiar Singular Value Decomposition (SVD), but they transcend the limitations of matrices (second-order tensors). SVD is a powerful tool that has achieved impressive results in information retrieval, collaborative filtering, computational linguistics, computational vision, and other fields. However, SVD is limited to two-dimensional arrays of data (two modes), and many potential applications have three or more modes, which require higher-order tensor decompositions. This paper evaluates four algorithms for higher-order tensor decomposition: Higher-Order Singular Value Decomposition (HO-SVD), Higher-Order Orthogonal Iteration (HOOI), Slice Projection (SP), and Multislice Projection (MP). We measure the time (elapsed run time), space (RAM and disk space requirements), and fit (tensor reconstruction accuracy) of the four algorithms, under a variety of conditions. We find that standard implementations of HO-SVD and HOOI do not scale up to larger tensors, due to increasing RAM requirements. We recommend HOOI for tensors that are small enough for the available RAM and MP for larger tensors

Robust L1-norm Singular-Value Decomposition and Estimation

Singular-Value Decomposition (SVD) is a ubiquitous data analysis method in engineering, science, and statistics. Singular-value estimation, in particular, is of critical importance in an array of engineering applications, such as channel estimation in communication systems, EMG signal analysis, and image compression, to name just a few. Conventional SVD of a data matrix coincides with standard Principal-Component Analysis (PCA). The L2-norm (sum of squared values) formulation of PCA promotes peripheral data points and, thus, makes PCA sensitive against outliers. Naturally, SVD inherits this outlier sensitivity. In this work, we present a novel robust method for SVD based on a L1-norm (sum of absolute values) formulation, namely L1-norm compact Singular-Value Decomposition (L1-cSVD). We then propose a closed-form algorithm to solve this problem and find the robust singular values with cost $\mathcal{O}(N^3K^2)$. Accordingly, the proposed method demonstrates sturdy resistance against outliers, especially for singular values estimation, and can facilitate more reliable data analysis and processing in a wide range of engineering applications

Enhanced Singular Value Decomposition based Fusion for Super Resolution Image Reconstruction

The singular value decomposition (SVD) plays a very important role in the field of image processing for applications such as feature extraction, image compression, etc. The main objective is to enhance the resolution of the image based on Singular Value Decomposition. The original image and the subsequent sub-pixel shifted image, subjected to image registration is transferred to SVD domain. An enhanced method of choosing the singular values from the SVD domain images to reconstruct a high resolution image using fusion techniques is proposesed. This technique is called as enhanced SVD based fusion. Significant improvement in the performance is observed by applying enhanced SVD method preceding the various interpolation methods which are incorporated. The technique has high advantage and computationally fast which is most needed for satellite imaging, high definition television broadcasting, medical imaging diagnosis, military surveillance, remote sensing etc

Singular Value Decomposition and Entropy Dimension of Fractals

We analyze the singular value decomposition (SVD) and SVD entropy of Cantor fractals produced by the Kronecker product. Our primary results show that SVD entropy is a measure of image ``complexity dimension" that is invariant under the number of Kronecker-product self-iterations (i.e., fractal order). SVD entropy is therefore similar to the fractal Hausdorff complexity dimension but suitable for characterizing fractal wave phenomena. Our field-based normalization (Renyi entropy index = 1) illustrates the uncommon step-shaped and cluster-patterned distributions of the fractal singular values and their SVD entropy. As a modal measure of complexity, SVD entropy has uses for a variety of wireless communication, free-space optical, and remote sensing applications