Approximate Orthogonality Research Articles

Robust wideband beamforming algorithm based on frequency difference technique

This paper investigates a novel robust wideband beamforming algorithm based on the frequency-difference (FD) technique. While FD technique can maintain good frequency invariance (FI) of the beamformer, it introduces cross terms that can adversely affect performance. To reduce the cross terms, a projection preprocessing is implemented prior to FD operation exploiting the approximate orthogonality between steering vectors (SVs). Then, a robust beamformer is designed based on the estimated cross terms covariance matrix (CTCM) and SV of the signal of interest (SOI). The output of beamformer is an FD SOI. To restore the SOI, we design a filtering system and apply a simple diagonal loading beamforming to obtain the amplitude and phase of SOI, respectively. The simulation results have verified the effectiveness and superiority of the proposed algorithm in the presence of array errors.

Non-negative Tucker decomposition with double constraints for multiway dimensionality reduction

<p>Nonnegative Tucker decomposition (NTD) is one of the renowned techniques in feature extraction and representation for nonnegative high-dimensional tensor data. The main focus behind the NTD-like model was how to factorize the data to get ahold of a high quality data representation from multidimensional directions. However, existing NTD-like models do not consider relationship and properties between the factor matrix of columns while preserving the geometric structure of the data space. In this paper, we managed to capture nonlinear local features of data space and further enhance expressiveness of the NTD clustering method by syncretizing organically approximately orthogonal constraint and graph regularized constraint. First, based on the uni-side and bi-side approximate orthogonality, we flexibly proposed two novel approximately orthogonal NTD with graph regularized models, which not only in part make the factor matrix tend to be orthogonality, but also preserve the geometrical information from high-dimensional tensor data. Second, we developed the iterative updating algorithm dependent on the multiplicative update rule to solve the proposed models, and provided its convergence and computational complexity. Finally, we used numerical experimental results to demonstrate the effectiveness, robustness, and efficiency of the proposed new methods on the real-world image datasets.</p>

Approximate orthogonality of permutation operators, with application to quantum information

Approximate orthogonality of permutation operators, with application to quantum information

Convex functions and approximate Birkhoff–James orthogonality

AbstractFor the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.

Some study of the approximate Birkhoff orthogonality and orthogonality of bounded linear operators

Some study of the approximate Birkhoff orthogonality and orthogonality of bounded linear operators

Direction Finding in Partly Calibrated Arrays Exploiting the Whole Array Aperture

We consider the problem of direction finding using partly calibrated arrays, a distributed subarray with position errors between subarrays. The key challenge is to enhance angular resolution in the presence of position errors. To achieve this goal, existing algorithms, such as subspace separation and sparse recovery, have to rely on multiple snapshots, which increases the burden of data transmission and the processing delay. Therefore, we aim to enhance angular resolution using only a single snapshot. To this end, we exploit the orthogonality of the signals of partly calibrated arrays. Particularly, we transform the signal model into a multiple-measurement model, and show that there is approximate orthogonality between the source signals in this model. We then use blind source separation to exploit the orthogonality. Simulation and experiment results both verify that our proposed algorithm achieves high angular resolution as distributed arrays without position errors, inversely proportional to the whole array aperture.

Micro-Doppler Extraction for Cone-Shaped Target Using Constrained Nonnegative Matrix Factorization

Micro-Doppler (MD) extraction is the precondition for target classification based on micro-motion. This letter proposes a method for MD extraction based on constrained nonnegative matrix factorization (NMF). The proposed algorithm is applied to the time-frequency spectrogram of the cone-shaped precession target. Taking the sparsity constraint, temporal continuity constraint and approximate orthogonality constraint into consideration, the basic NMF is improved to match the MD separation problem, and thus the time-frequency spectrogram of each scattering center is reconstructed by deriving the basis matrix and coefficient matrix. Morphological processing, valid frequency extraction and MD curve fitting are conducted on the reconstructed spectrograms to extract the MD accurately. Experiment results show that the proposed approach outperforms other algorithms with better accuracy and robustness.

Approximate Birkhoff orthogonalities in normed spaces

In a normed space we characterize the approximate orthogonality in terms of extreme points of the closed unit ball in dual space. There is also another (and older) definition of an approximate orthogonality and therefore in this paper both approximate orthogonalities are compared. Moreover, we characterize the approximate orthogonality in the class of compact operators on a reflexive Banach space.

Randomized Gram--Schmidt Process with Application to GMRES

A randomized Gram--Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical Gram--Schmidt process while being at least as numerically stable as the modified Gram--Schmidt process. Our approach is based on random sketching, which is a dimension reduction technique consisting in estimation of inner products of high-dimensional vectors by inner products of their small efficiently computable random images, so-called sketches. In this way, an approximate orthogonality of the full vectors can be obtained by orthogonalization of their sketches. The proposed Gram--Schmidt algorithm can provide computational cost reduction in any architecture. The benefit of random sketching can be amplified by performing the nondominant operations in higher precision. In this case the numerical stability can be guaranteed with a working unit roundoff independent of the dimension of the problem. The proposed Gram--Schmidt process can be applied to Arnoldi iteration and results in new Krylov subspace methods for solving high-dimensional systems of equations or eigenvalue problems. Among them we chose the randomized GMRES method as a practical application of the methodology.

A New Approximate Birkhoff Orthogonality Type

In this note, we introduce a new approximate Birkhoff orthogonality type and give a characterization for inner product spaces using the approximate orthogonality. We show some general properties of the approximate Birkhoff orthogonality type as well as applications. In particular, we study the relationship between the new approximate Birkhoff orthogonality type and other approximate orthogonality types that have been defined before. Furthermore we study the approximate preserving mapping and give some properties.

Denoising method in fiber optic current transformer based on data characteristics and depth entropy

Focusing on the difficulty in suppressing the random error of non-Gaussian noise in fiber optic current transformer (FOCT) measurement, a FOCT noise suppression method based on data modal characteristics and depth entropy is proposed. The empirical mode decomposition model is established, and the signal timing is processed with adaptability, completeness and approximate orthogonality. The measurement signals with high-amplitude interference distributed in each IMF are extracted to solve the problem of modal aliasing interference. The analysis model of the random dynamic characteristics of the time series is established, and the calculation method of the depth entropy is proposed. By calculating the correlation of the time series, the complexity of the information is evaluated, and the signal after filtering the interference is reconstructed. Experiment results show that the average value of the filtered current data is 0.2% deviation from the reference current, and the variance is reduced by 56%. This filtering method can significantly reduce the dispersion of test data while ensuring unbiased estimation. The​ FOCT noise suppression method based on data modal characteristics and depth entropy can filter non-Gaussian noise and the signal-to-noise ratio of FOCT output data can be improved.

Approximate orthogonality in complex normed spaces: a directional approach

We introduce and study the notion of approximate directional orthogonality in a complex normed space. We further explore the relevant scenario in the topological setting, which allows us to characterize approximate Birkhoff–James orthogonality of operators on a finite-dimensional complex Banach space. Our investigation extends some existing results in the literature to a broader scope.

Properties of Chmielinski-orthogonality using Kadets-Klee property

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

Distance formulae and best approximation in the space of compact operators

We present several distance formulae in the space of compact operators in terms of extreme points and semi-inner-products. We characterize best approximation to an element out of a subspace and obtain a sufficient condition for unique best approximation. The usefulness of the results is demonstrated with examples. Finally, using the distance formulae we characterize approximate Birkhoff-James orthogonality of an element to a subspace in the space of compact operators.

Approximate orthogonality in normed spaces and its applications II

In a normed space we consider an approximate orthogonality relation related to the Birkhoff orthogonality ⊥B. We prove thatx⊥Bεy⇔∃z∈Lin{x,y}x⊥Bz,‖z−y‖≤ε‖y‖⇔∃φ∈J(x)|φ(y)|≤ε‖y‖. Such a characterization was already proved in Chmieliński et al. (2017) [5] for real normed spaces. This result is a useful theoretical tool that was used in many papers. Therefore, we prove it now for complex normed spaces, as well as give some new applications. In particular, we extend the Arambašić-Rajić Theorem. Moreover, this paper presents some result concerning the problem of minimal extensions of continuous linear operators from hyperplanes; namely, some estimation for λA(Y,X) is proved.

On approximate orthogonality and symmetry of operators in semi-Hilbertian structure

The purpose of the article is to generalize the concept of approximate Birkhoff-James orthogonality, in the semi-Hilbertian structure. Given a positive operator A on a Hilbert space H, we define (ϵ,A)-approximate orthogonality and (ϵ,A)-approximate orthogonality in the sense of Chmieliński and establish a relation between them. We also characterize (ϵ,A)-approximate orthogonality in the sense of Chmieliński for A-bounded and A-bounded compact operators. We further generalize the concept of right symmetric and left symmetric operators on a Hilbert space. The utility of these notions is illustrated by extending some of the previous results obtained by various authors in the setting of Hilbert spaces.

Some remarks on the generalization of orthogonality in terms of operators

This paper deals with a generalization of orthogonality in terms of bounded linear operators on a Banach space. The goal is to find a relation between orthogonality of images and orthogonality of elements. We prove that if the images of a bounded linear operator are orthogonal in the Pythagorean sense, then the elements are orthogonal in the sense of Birkhoff's definition. In the case of Robert's orthogonality in terms of bounded linear operators under the restriction that any element belongs to the intersection of the norm attainment set of $T_1+\lambda T_2$ and $T_1-\lambda T_2$, if the images are orthogonal, then it implies that the operators are also orthogonal. Furthermore, some results in relation to the Carlsson, isosceles, and approximate Birkhoff-James orthogonality have been obtained. <br><br> У цій роботі розглядяється узагальнення ортогональності в термінах обмежених лінійних операторів на банаховому просторі. Метою роботи є знайти співвідношення між ортогональністю зображень і ортогональністю елементів. Доведено, що якщо образи обмеженого лінійного оператора ортогональні в піфагоровому сенсі, то елементи є ортогональний у сенсі визначення Біркгофа. У випадку ортогональності Роберта в термінах обмежених лінійних операторів із умовою, що будь-який елемент належить до перетину множин де оператори $T_1+\lambda T_2$ і $T_1-\lambda T_2$ досягають норми, з ортогональності образів випливає, що оператори є також ортогональні. Також отримано деякі результати про ортогональність в сенсі Карлссона, рівнобічної ортогональності та наближеної ортогональності в сенсі Біркгофа-Джеймса.

On a rho -Orthogonally Additive Mappings

We show that a real normed linear space endowed with the rho -orthogonality relation, in general need not be an orthogonality space in the sense of Rätz. However, we prove that rho -orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.

Approximate $\rho _{\lambda }^{v}$ -orthogonality and its preservation

We introduce and study a generalized approximate orthogonality relation in real normed linear spaces, namely, approximate ρλv-orthogonality. We investigate the relation between this generalized approximate orthogonality and approximate Birkhoff–James orthogonality. In particular, we show that every approximately ρλv-orthogonality-preserving linear mapping is necessarily a scalar multiple of an almost isometry.

Approximate orthogonality of complex fuzzy sets and approximately orthogonality preserving operators

Approximate orthogonality of complex fuzzy sets and approximately orthogonality preserving operators