Concentration of norms of random vectors with independent p-sub-exponential coordinates

CHAPTER 2 - Norms of Vectors and Matrices

The objective of maximum stiffness problem is to minimize the displacement. For this purpose, an optimization analysis is needed in order to obtain the structural shape with maximum stiffness. Displacement for structures with multiple degrees of freedom which has been modeled using finite element method is expressed in the form of a vector. In order to measure the size of vector quantity, a vector norm is used. The use of different types of vector norms might yield different results in maximum stiffness problem with shape as the design variable. This paper studies the effect of types of vector norms used in the outcome of analysis in maximum stiffness problem. Three types of vector norms, namely Euclidean, absolute value and maximum value norms, are studied. The results show that Euclidean norm is the most effective norm to be used.

The need for real-time computation of the Euclidean norm of a vector arises frequently in many signal processing applications such as vector median filtering, vector quantization and multiple-input multiple-output wireless communication systems. In this correspondence, we examine the properties of a linear combination of the 1-norm and the infinity norm as an approximation to the Euclidean norm of real-valued vectors. The approximation requires only two multiplications regardless of the vector length and does not require sorting of the absolute values of the vector entries. Numerical results show that the considered approximation incurs negligible performance degradations in typical applications.

Let ź · ź be the euclidean norm on Rn and let źn be the (standard) Gaussian measure on Rn with density $$(2\pi )^{ - n/2} e^{ - \left\| x \right\|^2 /2} $$ . Let ź (ź 1.3489795) be defined by $$\gamma _1 ([ - \vartheta /2, \vartheta /2]) = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $$ and let L be a lattice in Rn generated by vectors of norm ≤ ź. Then, for any closed convex set V in Rn with $$\gamma _n (V) \ge \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $$ , we have L + V = Rn (equivalently, for any a ź Rn, (a + L) ź V ź ź). The above statement can also be viewed as a nonsymmetric version of the Minkowski theorem.

S ATELLITE orbital transfer has been studied based on two main approaches: impulsive maneuvers and continuous-thrust maneuvers. Impulsive models, in spite of being approximate, are important because they enable mission designers to obtain a preliminary fuel budget as well as maneuver scheduling. Coplanar maneuvers usually consume less propellant than more general transfers wherein the orbital planes are modified. In this context, bi-impulsive maneuvers, such as the Hohmann transfer, are useful because they require a minimum number of thruster firings for each transfer between nonintersecting orbits. Under certain conditions, Hohmann transfers are optimal in the sense of minimizing the total velocity change. However, Hohmann transfers are designed to transfer a satellite between two collinear position vectors. When the initial and final position vectors are not collinear, one should consider other strategies. Several authors have studied the problem of optimal impulsive transfers. Munick et al. [1] provided a solution for optimal transfers from elliptic orbits to coplanar nonintersecting circular orbits using two velocity impulses. Lawden [2] derived a set of algebraic equations whose solution yields the optimal bi-impulsive transfer between two coplanar elliptic orbits. The obtained set of equations must be numerically solved. Battin [3] wrote an ‘2 cost function for a bi-impulsive transfer between twopositionvectors, assuming that the initial and final position and velocity vectors are all coplanar. The minimization of this cost function relies on numerical methods, since an eight-order algebraic equation must be solved. Battin’s development leads to necessary conditions, but no closed-form expressions for the optimal solution are provided. Broucke and Prado [4] studied the optimal coplanar transfer problem with two, three, and four impulses. They also obtained sets of algebraic equations that need to be numerically solved. Won [5] derived analytical conditions for the bi-impulsive transfer between elliptic coplanar coaxial orbits that minimize either the required velocity change or the transfer time. Ma et al. [6] developed an approach using the eccentricity vector, and derived equations to obtain the optimal bi-impulsive transfer between coplanar orbits. As in the previouslymentioned cases, the equationsmust be numerically solved.Conway and Pontani [7] derived optimal four-impulse orbital rendezvous maneuvers when both the chaser and target are in the same circular orbit with an initial phase difference of rad. They obtained general expressions for the change of velocity required by each impulse, and numerically optimized the total velocity change. An implicit assumption in most of the aforementioned works is that the propellant consumption is measured by the Euclidean norm of the velocity change vectors. This is appropriate if the satellite uses a single thruster to perform themaneuver, i.e., it regulates the attitude to align the nozzle with the required vector thrust. However, as also noted by others [8], in satellites that use multiple body-fixed orthogonal thrusters the mass consumption is measured by the ‘1-norm of the thrust vector rather than Euclidean norm. Thus, if Euclidean norms were used to find the fuel-optimal transfers for satellites equipped with body-fixed orthogonal thrusters (e.g., the Mango satellite in the PRISMA mission [9]), a nonfuel-optimal maneuver would be determined. Therefore, for satellites using orthogonal thrusters the optimization approach should be modified. With this consideration in mind, this work formulates the ‘1 cost function for a general coplanar transfer between two noncollinear position vectors with prescribed terminal velocities, and analytically determines the set of points that satisfy the necessary conditions for minima. Since this set includes only 14 points, it is easy to evaluate all of theminima candidates and obtain the globally optimal bi-impulsive maneuver for the mentioned scenario, avoiding numerical optimization processes. The derived analytical solution enables real-time onboard implementation.

Asymptotics of the norm of elliptical random vectors

We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. d-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since the diameter is a max $U$-statistic, hence the maximum of dependent random variables. Therefore, the limiting distributions may not be extreme value distributions. We obtain exhaustive results for the Euclidean diameter of a cloud of elliptical vectors whose Euclidean norm is in the domain of attraction for the maximum of the Gumbel distribution. We also obtain results in other norms for spherical vectors and we give several bi-dimensional generalizations. The main idea behind our results and their proofs is a specific property of random vectors whose norm is in the domain of attraction of the Gumbel distribution: the localization into subspaces of low dimension of vectors with a large norm.

Vector attenuation bias in the classical errors-in-variables model

On the MMSE estimation of norm of a Gaussian vector under additive white Gaussian noise with randomly missing input entries

We consider the problems of computing the Euclidean norm of the difference of two vectors and, as an application, computing the large components (Heavy Hitters) in the difference. We provide protocols that are approximate but private in the semi-honest model and efficient in terms of time and communication in the vector length N. We provide the following, which can serve as building blocks to other protocols: Euclidean norm problem: we give a protocol with quasi-linear local computation and polylogarithmic communication in N leaking only the true value of the norm. For processing massive datasets, the intended application, where N is typically huge, our improvement over a recent result with quadratic runtime is significant. Heavy Hitters problem: suppose, for a prescribed B, we want the B largest components in the difference vector. We give a protocol with quasi-linear local computation and polylogarithmic communication leaking only the set of true B largest components and the Euclidean norm of the difference vector. We justify the leakage as (1) desirable, since it gives a measure of goodness of approximation; or (2) inevitable, since we show that there are contexts where linear communication is required for approximating the Heavy Hitters. KeywordsNorm EstimationEuclidean NormPseudorandom GeneratorOblivious TransferPrivate Information RetrievalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Multilayer perceptrons (MLP) are often trained by minimizing the mean of squared errors (MSE), which is a sum of squared Euclidean norms of error vectors. Less common is to minimize the sum of Euclidean norms without squaring them. The latter approach, mean of non-squared errors (ME), bears implications from robust statistics. We carried out computational experiments to see if it would be notably better to train an MLP classifier by minimizing ME instead of MSE in the special case when training data contains class noise, i.e., when there is some mislabeling. Based on our experiments, we conclude that for small datasets containing class noise, ME could indeed be a very preferable choice, whereas for larger datasets it may not help.

We consider the computation of the Euclidean (or L2) norm of an n -dimensional vector in floating-point arithmetic. We review the classical solutions used to avoid spurious overflow or underflow and/or to obtain very accurate results. We modify a recently published algorithm (that uses double-word arithmetic) to allow for a very accurate solution, free of spurious overflows and underflows. To that purpose, we use a double-word square-root algorithm of which we provide a tight error analysis. The returned L2 norm will be within very slightly more than 0.5 ulp from the exact result, which means that we will almost always provide correct rounding.

High performance magnets play an important role in critical issues of modern life such as renewable energy supply, independence of fossile resource and electro mobility. The performance optimization of the established magnetic material system relies mostly on the microstructure control and modification. Here, finite element based in‐silico characterizations, as micromagnetic simulations can be used to predict the magnetization distribution on fine scales. The evolution of the magnetization vectors is described within the framework of the micromagnetic theory by the Landau‐Lifshitz‐Gilbert equation, which requires the numerically challenging preservation of the Euclidean norm of the magnetization vectors. Finite elements have proven to be particularly suitable for an accurate discretization of complex microstructures. However, when introducing the magnetization vectors in terms of a cartesian coordinate system, finite elements do not preserve their unit length a priori. Hence, additional numerical methods have to be considered to fulfill this requirement. This work introduces a perturbed Lagrangian multiplier to penalize all deviations of the magnetization vectors from the Euclidean norm in a suited manner. To reduce the resulting system of equations, an element level based condensation of the Lagrangian multiplier is presented.

A digit-recurrence algorithm for computing the Euclidean norm of a three-dimensional (3D) vector which often appears in 3D computer graphics is proposed. One of the three squarings required for the usual computation is removed and the other two squarings, as well as the two additions, are overlapped with the square rooting. The Euclidean norm is computed by iteration of carry-propagation-free additions, shifts, and multiplications by one digit. Different specific versions of the algorithm are possible, depending on the radix, the redundancy factor of the digit set, and etc. Each version of the algorithm can be implemented as a sequential (folded) circuit or a combinational (unfolded) circuit, which has a regular array structure suitable for VLSI.

This paper concerns the robust regression model when the number of predictors and the number of observations grow in a similar rate. Theory for M-estimators in this regime has been recently developed by several authors (El Karoui et al., 2013; Bean et al., 2013; Donoho and Montanari, 2013). Motivated by the inability of M-estimators to successfully estimate the Euclidean norm of the coefficient vector, we consider a Bayesian framework for this model. We suggest a two-component mixture of normals prior for the coefficients and develop a Gibbs sampler procedure for sampling from relevant posterior distributions, while utilizing a scale mixture of normal representation for the error distribution. Unlike M-estimators, the proposed Bayes estimator is consistent in the Euclidean norm sense. Simulation results demonstrate the superiority of the Bayes estimator over traditional estimation methods.