Operator Norm Research Articles

Covariance estimation under one-bit quantization

We consider the classical problem of estimating the covariance matrix of a sub-Gaussian distribution from i.i.d. samples in the novel context of coarse quantization, that is, instead of having full knowledge of the samples, they are quantized to one or two bits per entry. This problem occurs naturally in signal processing applications. We introduce new estimators in two different quantization scenarios and derive nonasymptotic estimation error bounds in terms of the operator norm. In the first scenario, we consider a simple, scale-invariant one-bit quantizer and derive an estimation result for the correlation matrix of a centered Gaussian distribution. In the second scenario, we add random dithering to the quantizer. In this case, we can accurately estimate the full covariance matrix of a general sub-Gaussian distribution by collecting two bits per entry of each sample. In both scenarios, our bounds apply to masked covariance estimation. We demonstrate the near optimality of our error bounds by deriving corresponding (minimax) lower bounds and using numerical simulations.

Tight computationally efficient approximation of matrix norms with applications

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of ``solvable cases,'' most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (``ellitopic'' and ``co-ellitopic,'' respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding ``robust operator norm of uncertain matrix with box uncertainty,'' that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\leq1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of ``uncertain-but-bounded'' noise varying in a box.

Equivalence condition for the best matching parameters of multiple integral operator with generalized homogeneous kernel and applications

By introducing the matching parameters $a_{1},a_{2},\ldots,a_{n}$ and using the weight function method, the inequality $~\|T(f_{1},f_{2},\ldots,f_{n-1})\|_{q_{n},\alpha_{n}(1-q_{n})}\!\leq\!~M(a_{1},a_{2},\ldots,a_{n})~\prod_{i=1}^{n-1}\|f_{i}\|_{p_{i},\alpha_{i}}$ can be obtained for the operator $T$: <disp-formula><tex-math><![CDATA[$$ T(f_{1},f_{2},\ldots,f_{n-1})(x_{n})=\int_{\mathbb{R}_{+}^{n-1}}K(x_{1},\ldots,x_{n-1},x_{n})\prod_{i=1}^{n-1}f_{i}(x_{i}) {d}x_{1}\cdots{d}x_{n-1}.$$]]></tex-math></disp-formula> If the constant factor $M(a_{1},a_{2},\ldots,a_{n})$ is the operator norm $\|T\|$ of $T:\prod_{i=1}^{n-1}L_{p_{i}}^{\alpha_{i}}(0,+\infty)\rightarrow~L_{q_{n}}^{\alpha_{n}(1-q_{n})}(0,+\infty)$, then $a_{1},a_{2},\ldots,a_{n}$ are called the best matching parameters of the operator $T$. In this paper, for the generalized homogeneous kernel, the necessary and sufficient conditions for the best matching parameters of the operator $T$ are discussed, as the method of determining that the operator $T$ is bounded and the calculation formula for the operator norm of $T$ are obtained. Finally, some special cases are given as applications.

The role of leadership in educational innovation: a comparison of two mathematics departments’ initiation, implementation, and sustainment of active learning

Several studies have shown that the use of active learning strategies can help improve student success and persistence in STEM-related fields. Despite this, widespread adoption of active learning strategies is not yet a reality as institutional change can be difficult to enact. Accordingly, it is important to understand how departments in institutions of higher education can initiate and sustain meaningful change. We use interview data collected from two institutions to examine how leaders at two universities contributed to the initiation, implementation, and sustainability of active learning in undergraduate calculus and precalculus courses. At each institution, we spoke to 27 stakeholders involved in changes (including administrators, department chairs, course coordinators, instructors, and students). Our results show that the success of these changes rested on the ability of leaders to stimulate significant cultural shifts within the mathematics department. We use communities of transformation theory and the four-frame model of organization change in STEM departments in order to better understand how leaders enabled such cultural shifts. Our study highlights actions leaders may take to support efforts at improving education by normalizing the use of active learning strategies and provides potential reasons for the efficacy of such actions. These results underscore the importance of establishing flexible, distributed leadership models that attend to the cultural and operational norms of a department. Such results may inform leaders at other institutions looking to improve education in their STEM departments.

Norms of Basic Operators in Vector Valued Model Spaces and de Branges Spaces

Let \(\Omega _+\) be either the open unit disc or the open upper half plane or the open right half plane. In this paper, we compute the norm of the basic operator \(A_\alpha =\Pi _\Theta T_{b_\alpha }|_{\mathcal {H}(\Theta )}\) in the vector valued model space \(\mathcal {H}(\Theta )=H^m_2 \ominus \Theta H^m_2\) associated with an \(m\times m\) matrix valued inner function \(\Theta \) in \(\Omega _+\) and show that the norm is attained. Here \(\Pi _\Theta \) denotes the orthogonal projection from the Lebesgue space \(L^m_2\) onto \(\mathcal {H}(\Theta )\) and \(T_{b_\alpha }\) is the operator of multiplication by the elementary Blaschke factor \(b_{\alpha }\) of degree one with a zero at a point \(\alpha \in \Omega _+\). We show that if \(A_\alpha \) is strictly contractive, then its norm may be expressed in terms of the singular values of \(\Theta (\alpha )\). We then extend this evaluation to the more general setting of vector valued de Branges spaces.

Bounding imprecise failure probabilities in structural mechanics based on maximum standard deviation

This paper proposes a framework to calculate the bounds on failure probability of linear structural systems whose performance is affected by both random variables and interval variables. This kind of problems is known to be very challenging, as it demands coping with aleatoric and epistemic uncertainty explicitly. Inspired by the framework of the operator norm theorem, it is proposed to consider the maximum standard deviation of the structural response as a proxy for detecting the crisp values of the interval parameters, which yield the bounds of the failure probability. The scope of application of the proposed approach comprises linear structural systems, whose properties may be affected by both aleatoric and epistemic uncertainty and that are subjected to (possibly imprecise) Gaussian loading. Numerical examples indicate that the application of such proxy leads to substantial numerical advantages when compared to a traditional double-loop approach for coping with imprecise failure probabilities. In fact, the proposed framework allows to decouple the propagation of aleatoric and epistemic uncertainty.

Further norm inequalities for sums of operators

We give several further norm inequalities for sums of bounded linear operators on a complex Hilbert space. These inequalities refine and generalize earlier inequalities. Among other applications, it is shown that for A∈B(H) and r∈[0,1],w(A)≤12‖A‖+12‖|A|r|A⁎|r‖12‖|A|1−r|A⁎|1−r‖12, which refines a result of Kittaneh (2003) [5]. Here, w(⋅) and ‖⋅‖ denote the numerical radius and the usual operator norm, respectively.

Norm and numerical radius inequalities for Hilbert space operators

Norm and numerical radius inequalities for Hilbert space operators

Matrix means and a novel high-dimensional shrinkage phenomenon

Many statistical settings call for estimating a population parameter, most typically the population mean, based on a sample of matrices. The most natural estimate of the population mean is the arithmetic mean, but there are many other matrix means that may behave differently, especially in high dimensions. Here we consider the matrix harmonic mean as an alternative to the arithmetic matrix mean. We show that in certain high-dimensional regimes, the harmonic mean yields an improvement over the arithmetic mean in estimation error as measured by the operator norm. Counter-intuitively, studying the asymptotic behavior of these two matrix means in a spiked covariance estimation problem, we find that this improvement in operator norm error does not imply better recovery of the leading eigenvector. We also show that a Rao-Blackwellized version of the harmonic mean is equivalent to a linear shrinkage estimator studied previously in the high-dimensional covariance estimation literature, while applying a similar Rao-Blackwellization to regularized sample covariance matrices yields a novel nonlinear shrinkage estimator. Simulations complement the theoretical results, illustrating the conditions under which the harmonic matrix mean yields an empirically better estimate.

High-dimensional variable screening through kernel-based conditional mean dependence

This article proposes a model-free feature screening method for ultrahigh-dimensional data. The proposed method is built on the called kernel-based conditional mean dependence, which is defined based on the reproducing kernel embedding and the Hilbert–Schmidt norm of a tensor operator. Theoretically, both sure screening and rank consistency properties are established under weak assumptions. Through an extensive simulation study as well as a real data analysis, it is illustrated that the statistics generated by other kernels in distance kernel family are more sensitive to feature screening in ultra-high dimensions.

Dimension-free bounds and structural results in communication complexity

The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to perhaps the more common framework in communication complexity where poly-logarithmic dependencies on the number of input bits are tolerated.Dimension-free bounds are also closely related to structural results, where one seeks to describe the structure of Boolean matrices and functions that have low complexity. We prove such theorems for several communication and query complexity measures as well as various matrix and operator norms. In several other cases we show that such bounds do not exist.We propose several conjectures, and establish that, in addition to applications in complexity theory, these problems are central to characterization of the idempotents of the algebra of Schur multipliers, and could lead to new extensions of Cohen’s celebrated idempotent theorem regarding the Fourier algebra.

A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems

Abstract Using a dynamical step size technique, a new self-adaptive CQ-algorithm is proposed in the presence of an inertial term to find the solution of convex feasibility problem and monotone inclusion problem involving a finite number of maximal monotone set valued operators. To do this, in certain Banach spaces, we construct an algorithm which converges to the fixed point of right Bregman strongly nonexpansive mappings and coincidentally solves the convex feasibility and monotone inclusion problems. Strong convergence of the algorithm is achieved without computation of the associated operator norms. Interesting numerical examples which illustrate the implementation and efficiency of our scheme are also given. Results obtained via this work improve and extend on previous results of its kind, in the literature.

Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems

The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretised Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver—and also as a preconditioner—for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general domain decompositions) that ORAS is an unconventional finite element approximation of a classical parallel iterative Schwarz method, formulated at the PDE (non-discrete) level. This non-discrete Schwarz method was recently analysed in [Gong, Gander, Graham, Lafontaine, and Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation], and the present paper gives a corresponding discrete version of this analysis. In particular, for domain decompositions in strips in 2-d, we show that, when the mesh size is small enough, ORAS inherits the convergence properties of the Schwarz method, independent of polynomial order. The proof relies on characterising the ORAS iteration in terms of discrete ‘impedance-to-impedance maps’, which we prove (via a novel weighted finite-element error analysis) converge as h → 0 h\rightarrow 0 in the operator norm to their non-discrete counterparts.

Dirichlet-Morrey type spaces and Volterra integral operators

Let 0 < p < ∞ and K : [ 0 , ∞ ) → [ 0 , ∞ ) be a nondecreasing and right-continuous function. A new Dirichlet-Morrey type space D p − 1 p , K is introduced in this paper. The boundedness of the identity operator I d from D p − 1 p , K to tent spaces T K p ( μ ) is characterized. As an application, the boundedness, compactness and essential norm of Volterra integral operators on D p − 1 p , K are investigated.

Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

In this paper we extend the local iterative Lie-Schwinger block-diagonalization method – introduced in [8] for quantum lattice systems with bounded interactions in arbitrary dimension– to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t|independent of the size of the lattice, we prove that the spectral gap above the ground-state energy of such Hamiltonians remains strictly positive.As in [8], we iteratively construct a sequence of local block-diagonalization steps based on unitary conjugations of the original Hamiltonian and inspired by the Lie-Schwinger procedure. To control the supports of the effective potentials generated in the course of our block-diagonalization steps, we use methods introduced in [8] for Hamiltonians with bounded interactions potentials. However, due to the unboundedness of the interaction potentials, weighted operator norms must be introduced, and some of the steps of the inductive proof by which we control the weighted norms of the effective potentials require special care to cope with matrix elements of unbounded operators.We stress that no “large-field problems” appear in our construction. In this respect our operator methods turn out to be an efficient tool to separate the low-energy spectral region of the Hamiltonian from other spectral regions, where the unbounded nature of the interaction potentials would become manifest.

Approximations of Resolvents of Second Order Elliptic Operators with Periodic Coefficients

We consider second order elliptic operator with ε-periodic measurable coefficients in divergence form, acting in the space ℝd. For the resolvent of this operator we construct approximations in the operator norm $$ {\left\Vert \cdot \right\Vert}_{H^1\to {L}^2} $$ with the remainder of order ε3 as ε → 0.

Some new relations between the Berezin number and the Berezin norm of operators

Some new relations between the Berezin number and the Berezin norm of operators

The Relationship between Figureheads and Managerial Leaders in the Private University Sector: A Decentralised, Competency-Based Leadership Model for Sustainable Higher Education

Using school management settings, important substantial leadership models (such as “instructional”, “constructive”, and “distributed”) are developed. University leadership, especially in the private sector, is an understudied topic. While private universities need to generate revenue for profits and survival, they should at the same time retain a sound education philosophy—an extremely difficult task. Studies are yet to be conducted to understand how leaders of private universities handle this challenge, and this issue motivates the present study. Using Bangladesh as a case study, this qualitative analysis discovers that the “visionary” leadership approach is mainly used for the business management models that governs private-sector universities. This does not allow an established education leadership model to function well, if at all. This has seriously challenged education philosophy. Innovation of a specialised private-sector university leadership model is therefore required. A “distributed leadership” approach is the latest innovation for educational leadership, which is mainly applicable for the public system, especially for K12 provision. Considering a distinct operational norm of the private university sector in a developing nation, this research suggests a dynamic revenue-collection model which would enable the development of a decentralised, competency-based leadership approach. While the private university sector may not be able to implement “distributed leadership” model given the operational differences and challenges involved, the proposed model may support the private university sector to function in tertiary level, as the “distributed model” functions for K12 provision. This proposed model (decentralised, competency-based leadership) would further be able to ensure a specialised operational directive for private universities in an emerging nation that may ensure a distinctive nature for the sector’s delivery of sustainable higher education (HE) without compromising the philosophy and role of HE.

On the closure of absolutely norm attaining operators

ABSTRACTLet H1 and H2 be complex Hilbert spaces and T:H1→H2 be a bounded linear operator. We say T is norm attaining if there exists x∈H1 with ‖x‖=1 such that ‖Tx‖=‖T‖. If for every non-zero closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator or AN-operator. If we replace the norm of the operator by the minimum modulus m(T)=inf{‖Tx‖:x∈H1,‖x‖=1} in the above definitions, then T is called a minimum attaining and an absolutely minimum attaining operator or AM-operator, respectively. In this article, we discuss the operator norm closure of AN-operators. We completely characterize operators in this closure and study several important properties. We mainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for AM-operators and prove that the closure of AM-operators is the same as the closure of AN-operators. Consequently, we prove similar results for operators in the norm closure of AM-operators.

The norm of linear extension operators for [formula omitted

Fix integers m≥2, n≥1. We prove the existence of a bounded linear extension operator for Cm−1,1(Rn) with operator norm at most exp⁡(γDk), where D:=(m+n−1n) is the number of multiindices of length n and order at most m−1, and γ,k>0 are absolute constants (independent of m,n,E). Upper bounds on the norm of this operator are relevant to basic questions about fitting a smooth function to data. Our results improve on a previous construction of extension operators of norm at most exp⁡(γDk2D). Along the way, we establish a finiteness theorem for Cm−1,1(Rn) with improved bounds on the involved constants.