Approximate Orthogonality Research Articles

Convergence and quasi-optimality of an adaptive continuous interior multi-penalty finite element method

An adaptive continuous interior multi-penalty finite element method (ACMPFEM) for symmetric second-order linear elliptic equations is considered. The proposed ACMPFEM is defined using a bilinear form which not only penalize the jumps of the first order normal derivatives across the element edges but also the jumps of all normal derivatives up to order p. In addition, compared with the analyses for adaptive finite element method, extra works are done to overcome the difficulties caused by the additional penalty terms. To be precise, the first difficulty is that the variational formulation of the ACMPFEM is not consistent with the continuous problem for weak solution u just in and the second difficulty lies in the proof of the optimality of our ACMPFEM. Furthermore, to prove the convergence and quasi-optimality of the ACMPFEM, an approximate Galerkin orthogonality in which the penalty term is regarded as a perturbation, and estimating the perturbation very carefully is used and the definition of our approximation class has to be involved with the penalty terms. Numerical tests are provided to verify the theoretical results and show advantages of the ACMPFEM.

Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Let $ G $ be a connected, simply connected nilpotent Lie group and $ \Gamma < G $ a lattice. We prove that each ergodic diffeomorphism $ \phi(x\Gamma)=uA(x)\Gamma $ on the nilmanifold $ G/\Gamma $, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(\Gamma)=\Gamma $, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow: (i) Sarnak's conjecture on Mobius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Mobius orthogonality holds on so called typical short interval: $ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(\phi^n(x\Gamma))\mu (n)\right|\to 0$ as $ H\to\infty $ and $ H/M\to0 $ for each $ x\Gamma\in G/\Gamma $ and each $ f\in C(G/\Gamma) $. In particular, the results in (i) and (ii) hold for ergodic nil-translations. Moreover, we prove that each nilsequence is orthogonal to the Mobius function $\mu$ on a typical short interval. We also study the problem of lifting of the AOP property to induced actions and derive some applications on uniform distribution.

A strong robust observer of distributed drive electric vehicle states based on strong tracking-iterative central difference Kalman filter algorithm

This article proposed a strong robust observer of distributed drive electric vehicle states, including yaw rate, sideslip angle, and longitudinal velocity. Based on strong tracking filter algorithm framework, the proposed observer realized a strong tracking-iterative central difference Kalman filter by introducing a time-varying fade factor into iterative central difference Kalman filter. The introducing of time-varying fade factor assigns approximate orthogonality to residual error, which improves robustness of the observer at mutation conditions. Calculation efficiency and accuracy are improved by applying central difference transformation to approach posterior mean and posterior co-variance. By correcting variance and co-variance with combination of states updating and Gauss–Newton iteration, the observer also achieves high estimation accuracy and convergence rate. Finally, the observer was simulated in vehicle dynamics simulator veDYNA at slalom test and double lane change test with high and low road friction coefficients, respectively. Simulation results have verified that the observer has higher estimation accuracy as well as robustness comparing to extended Kalman filter.

Mappings preserving approximate orthogonality in Hilbert $C^*$-modules

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta , \varepsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta , \varepsilon )$-orthogonality preserving. In particular, if $\mathscr {E}$ is a full Hilbert $\mathscr {A}$-module with $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T, S\colon \mathscr {E}\to \mathscr {E}$ are two linear mappings satisfying $|\langle Sx, Sy\rangle | = \|S\|^2|\langle x, y\rangle |$ for all $x, y\in \mathscr {E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta , \varepsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T\colon \mathscr {E} \to \mathscr {F}$ is a nonzero $\mathscr {A}$-linear $(\delta , \varepsilon )$-orthogonality preserving mapping between $\mathscr {A}$-modules, then \[ \bigl \|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle \bigr \|\leq \frac {4(\varepsilon - \delta )}{(1 - \delta )(1 + \varepsilon )} \|Tx\|\|Ty\|\qquad (x, y\in \mathscr {E}). \] As a result, we present some characterizations of the orthogonality preserving mappings.

Local and Networked Mean-square Estimation with High Dimensional Log-concave Noise

We consider the problem of mean-square estimation of the state of a discrete time dynamical system having additive non-Gaussian noise. We assume that the noise has the structure that at each time instant, it is a projection of a fixed high-dimensional noise vector with a log-concave density. We derive conditions which guarantee that, as the dimension of this noise vector grows large, the optimal estimator of the problem with Gaussian noise becomes near-optimal for the problem with non-Gaussian noise. The results are derived by first showing an asymptotic Gaussian lower bound on the minimum error which holds even for nonlinear systems. The bound is asymptotically tight for linear systems. For nonlinear systems, this bound is tight provided and the noise has a strongly log-concave density with some additional structure. These results imply that the estimate obtained by employing the Gaussian estimator on the non-Gaussian problem satisfies an approximate orthogonality principle, and the difference between this estimate and the optimal estimate vanishes strongly in $L^{2}$ . For linear systems with high-dimensional log-concave noise of this structure, we get that the Kalman filter serves as a near-optimal estimator. A key ingredient in the proofs is a recent central limit theorem of Eldan and Klartag. The minimum mean-square error is in general neither weakly upper semicontinuous nor weakly lower semicontinuous. Our results show that in the setting we consider, it is weakly continuous. We then consider mean-square estimation over a noisy network, where the problem is to jointly design strategies for sensing and estimation to minimize a composite quadratic cost. These problems do not have a static information structure, whereby the previously derived results do not apply. We consider two different cases—when the estimation part of the problem corresponds to estimating the original source sensed by the sensor, and when this part corresponds to the estimation of the action of the sensor. In the former case, we show that results analogous to the above continue to hold. The latter case corresponds to a variant of Witsenhausen’s problem, wherein we show only partial results.

Characterizations of Operator Birkhoff–James Orthogonality

AbstractIn this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert C*-modules and certain elements of . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for we prove that if the norm attaining set is a unit sphere of some finite dimensional subspace of and , then for every , T is the strong Birkhoff–James orthogonal to S if and only if there exists a unit vector such that . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product C*-modules.

On the Output of Nonlinear Systems Excited by Discrete Prolate Spheroidal Sequences

The discrete prolate spheroidal sequences (DPSSs)—a set of optimally bandlimited sequences with unique properties—are important to applications in both science and engineering. In this work, properties of nonlinear system response due to DPSS excitation are reported. In particular, the sequences are shown to be approximately orthogonal after passing through a nonlinear, multiple-input multiple-output dynamical system under quite general conditions. This work quantifies these conditions in terms of constraints upon the higher-order generalized transfer functions characterizing the Volterra expansion of a MIMO system, the Volterra order of the system, and the DPSS bandwidth parameter $W$ and time-bandwidth parameter $NW$ . The approximate system output orthogonality allows multiple-input, multiple-output characterization of the linear narrowband response of a nonlinear system using simultaneous, DPSS excitation.

MIMO-SAR orthogonal waveform set design based on random subcarriers OFDM signal

The orthogonal waveform set with excellent characteristics is one of the key technologies to realize the advantages of the MIMO-SAR radar system. Among them, the orthogonality and Doppler tolerance are the most urgent requirements. The waveform sets with approximate rational orthogonality can be generated using interleaved subcarriers OFDM (I-OFDM) technique while maintaining the same range resolution as the OFDM signal, which has been paid more attention in MIMO radar waveform design. However, the narrow orthogonal time-delay range limits the design and application of I-OFDM orthogonal waveform sets. In this paper, random subcarriers OFDM (R-OFDM) design method is proposed. First, the cost function is constructed based on the PSLR and ISLR value of the autocorrelation and cross-correlation ambiguity function. Then, the subcarriers frequency distribution structure and the weighted scheme are obtained by the optimization algorithm. At last, the performance of the proposed waveform sets is verified by computer simulation.

Approximate orthogonality in normed spaces and its applications

In a normed space we consider an approximate orthogonality relation related to the Birkhoff orthogonality. We give some properties of this relation as well as applications. In particular, we characterize the approximate orthogonality in the class of linear bounded operators on a Hilbert space.

BIRKHOFF ORTHOGONALITY IN CLASSICAL -IDEALS

The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in ${\mathcal{L}}(X;Y)$ under the assumption that ${\mathcal{K}}(X;Y)$ is an $M$-ideal in ${\mathcal{L}}(X;Y)$. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in ${\mathcal{K}}(X;Y)$.

Orthogonality of compact operators

In this paper we characterize the Birkhoff–James orthogonality for elements of K(X;Y). In this way we extend the Bhatia–Šemrl theorem. As an application, we consider the approximate orthogonality preserving property. Moreover, we give a new characterization of inner product spaces.

Approximate Roberts orthogonality sets and $${(\delta, \varepsilon)}$$ ( δ , ε ) -(a, b)-isosceles-orthogonality preserving mappings

We introduce the concept of approximate Roberts orthogonality set and investigate the geometric properties of such sets. In addition, we introduce the notion of approximate a-isosceles-orthogonality and consider a class of mappings, which approximately preserve a-isosceles-orthogonality.

Linear mappings approximately preserving orthogonality in real normed spaces

In a normed space we introduce an exact and approximate orthogonality relation. We consider classes of linear mappings approximately preserving this kind of orthogonality. We show that, in particular, the property that a linear mapping approximately preserves the $B$-orthogonality is equivalent to that it approximately preserves the $\rho,\rho_+$-orthogonality (although these orthogonalities need not be equivalent). Moreover, we show that every approximately orthogonality preserving linear mapping is necessarily a scalar multiple of an almost isometry.

Higher rank numerical ranges of rectangular matrices

In this paper, the notions of rank$-k$ numerical range and $k-$spectrum of rectangular complex matrices are introduced. Some algebraic and geometrical properties are investigated. Moreover, for $\epsilon \gt 0,$ the notion of Birkhoff-James approximate orthogonality sets for $\epsilon$-higher rank numerical ranges of rectangular matrices is also introduced and studied. The proposed definitions yield a natural generalization of standard higher rank numerical ranges.

An Improved RIP-Based Performance Guarantee for Sparse Signal Recovery via Orthogonal Matching Pursuit

A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations is that the restricted isometry constant of the sensing matrix satisfies delta_K+1<1/(sqrt(delta_K+1)+1). By exploiting an approximate orthogonality condition characterized via the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the upper bound on delta can be further relaxed to delta_K+1<(sqrt(1+4*delta_K+1)-1)/(2K).This result thus narrows the gap between the so far best known bound and the ultimate performance guarantee delta_K+1<1/(sqrt(delta_K+1)) that is conjectured by Dai and Milenkovic in 2009. The proposed approximate orthogonality condition is also exploited to derive less restricted sufficient conditions for signal reconstruction in several compressive sensing problems, including signal recovery via OMP in a noisy environment, compressive domain interference cancellation, and support identification via the subspace pursuit algorithm.

Using instrumental variables to estimate a Cox’s proportional hazards regression subject to additive confounding

The estimation of treatment effects is one of the primary goals of statistics in medicine. Estimation based on observational studies is subject to confounding. Statistical methods for controlling bias due to confounding include regression adjustment, propensity scores and inverse probability weighted estimators. These methods require that all confounders are recorded in the data. The method of instrumental variables (IVs) can eliminate bias in observational studies even in the absence of information on confounders. We propose a method for integrating IVs within the framework of Cox's proportional hazards model and demonstrate the conditions under which it recovers the causal effect of treatment. The methodology is based on the approximate orthogonality of an instrument with unobserved confounders among those at risk. We derive an estimator as the solution to an estimating equation that resembles the score equation of the partial likelihood in much the same way as the traditional IV estimator resembles the normal equations. To justify this IV estimator for a Cox model we perform simulations to evaluate its operating characteristics. Finally, we apply the estimator to an observational study of the effect of coronary catheterization on survival.

The analysis of the simplification from the ideal ratio to binary mask in signal-to-noise ratio sense

For speech separation systems, the ideal binary mask (IBM) can be viewed as a simplified goal of the ideal ratio mask (IRM) which is derived from Wiener filter. The available research usually verify the rationality of this simplification from the aspect of speech intelligibility. However, the difference between the two masks has not been addressed rigorously in the signal-to-noise ratio (SNR) sense. In this paper, we analytically investigate the difference between the two ideal masks under the assumption of the approximate W-Disjoint Orthogonality (AWDO) which almost holds under many kinds of interference due to the sparse nature of speech. From the analysis, one theoretical upper bound of the difference is obtained under the AWDO assumption. Some other interesting discoveries include a new ratio mask which achieves higher SNR gains than the IRM and the essential relation between the AWDO degree and the SNR gain of the IRM.

Approximate Roberts orthogonality

In a real normed space we introduce two notions of approximate Roberts orthogonality as follows: $$x \perp_R^\varepsilon y \, {\rm if \, and \, only \, if} \left|\|x + ty\|^2 - \|x - ty\|^2\right| \leq 4\varepsilon\|x\|\|ty\| \, {\rm for \, all} \, t \in \mathbb{R}\,;$$ and $$x^{\varepsilon} \perp_R y \, {\rm if \, and \, only \, if} \left|\|x + ty\|-\|x - ty\|\right| \leq \varepsilon(\|x + ty\| + \|x - ty\|) \, {\rm for \, all} \, t \in \mathbb{R}\,.$$ We investigate their properties and their relationship with the approximate Birkhoff orthogonality. Moreover, we study the class of linear mappings preserving approximately Roberts orthogonality of type \({^{\varepsilon}\perp_R}\). A linear mapping \({U: \mathcal{X} \to \mathcal{Y}}\) between real normed spaces is called an \({\varepsilon}\)-isometry if \({(1 - \varphi_1 (\varepsilon))\|x\| \leq \|Ux\| \leq (1 + \varphi_2(\varepsilon))\|x\|\,\,(x \in \mathcal{X})}\), where \({\varphi_1 (\varepsilon)\rightarrow0}\) and \({\varphi_2 (\varepsilon)\rightarrow0}\) as \({\varepsilon\rightarrow 0}\). We show that a scalar multiple of an \({\varepsilon}\)-isometry is an approximately Roberts orthogonality preserving mapping.

The optimal ratio time-frequency mask for speech separation in terms of the signal-to-noise ratio

In this paper, a computational goal for a monaural speech separation system is proposed. Since this goal is derived by maximizing the signal-to-noise ratio (SNR), it is called the optimal ratio mask (ORM). Under the approximate W-Disjoint Orthogonality assumption which almost always holds due to the sparse nature of speech, theoretical analysis shows that the ORM can improve the SNR about 10log(10)2 dB over the ideal ratio mask. With three kinds of real-world interference, the speech separation results of SNR gain and objective quality evaluation demonstrate the correctness of the theoretical analysis, and imply that the ORM achieves a better separation performance.

Time frequency source separation and direction of arrival estimation in a 3D soundscape environment

The DUET (Degenerative Unmixing and Estimation Technique) algorithm has been shown to be an effective method for the separation of multiple sources from a stereo mixture. We present a time–frequency masking technique based on DUET for the blind separation of N sources from a four channel B format audio mixture. This method is applicable where sources that are both radially sparse in three dimensions, and exhibit approximate W-disjoint orthogonality, i.e. where the signals are approximately sparse in the time–frequency domain. Using the B-format mixture, we generate a three dimensional power-weighted geodesic histogram and show the peak locations correspond to the direction of arrival of the sources. These peaks are then used to generate a time–frequency mask to separate the sources from the w-channel of the B-format mixture. Experimental separation results using speech signals are presented.