Orthogonality

About: Orthogonality is a research topic. Over the lifetime, 3723 publications have been published within this topic receiving 59256 citations. The topic is also known as: orthogonal.

Orthogonal Tensor Decompositions

Abstract: We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or n-way arrays) using two different definitions of orthogonality. We present numerous examples to illustrate the difficulties in understanding such decompositions. We conclude with a counterexample to a tensor extension of the Eckart--Young SVD approximation theorem by Leibovici and Sabatier [Linear Algebra Appl., 269 (1998), pp. 307--329].

Orthogonal Column Latin Hypercubes and Their Application in Computer Experiments

Abstract: Latin hypercubes have been frequently used in conducting computer experiments. In this paper, a class of orthogonal Latin hypercubes that preserves orthogonality among columns is proposed. Applying an orthogonal Latin hypercube design to a computer experiment benefits the data analysis in two ways. First, it retains the orthogonality of traditional experimental designs. The estimates of linear effects of all factors are uncorrelated not only with each other, but also with the estimates of all quadratic effects and bilinear interactions. Second, it can facilitate nonparametric fitting procedures, because one can select good space-filling designs within the class of orthogonal Latin hypercubes according to selection criteria.

A dual-tree complex wavelet transform with improved orthogonality and symmetry properties

Abstract: We present a new form of the dual-tree complex wavelet transform (DT CWT) with improved orthogonality and symmetry properties. Beyond level 1, the previous form used alternate odd-length and even-length bi-orthogonal filter pairs in the two halves of the dual-tree, whereas the new form employs a single design of even-length filter with asymmetric coefficients. These are similar to the Daubechies orthonormal filters, but designed with the additional constraint that the filter group delay should be approximately one quarter of the sample period. The filters in the two trees are just the time-reverse of each other, as are the analysis and reconstruction filters. This leads to a transform, which can use shorter filters, which is orthonormal beyond level 1, and in which the two trees are very closely matched and have a more symmetric sub-sampling structure, but which preserves the key DT CWT advantages of approximate shift-invariance and good directional selectivity in multiple dimensions.

Optimality of high resolution array processing using the eigensystem approach

Abstract: In the classical approach to underwater passive listening, the medium is sampled in a convenient number of "look-directions" from which the signals are estimated in order to build an image of the noise field. In contrast, a modern trend is to consider the noise field as a global entity depending on few parameters to be estimated simultaneously. In a Gaussian context, it is worthwhile to consider the application of likelihood methods in order to derive a detection test for the number of sources and estimators for their locations and spectral levels. This paper aims to compute such estimators when the wavefront shapes are not assumed known a priori. This justifies results previously found using the asymptotical properties of the eigenvalue-eigenvector decomposition of the estimated spectral density matrix of the sensor signals: they have led to a variety of "high resolution" array processing methods. More specifically, a covariance matrix test for equality of the smallest eigenvalues is presented for source detection. For source localization, a "best fit" method and a test of orthogonality between the "smallest" eigenvectors and the "source" vectors are discussed.