Entries Of Matrix Research Articles

Matrices with hyperbolical Krein space numerical range

AbstractThis paper is devoted to matrices with hyperbolical Krein space numerical range. This shape characterizes the 2-by-2 case and persists for certain classes of matrices, independently of their size. Necessary and sufficient conditions for low dimensional tridiagonal matrices to have this shape are obtained involving only the matrix entries.

Local quadratic spectral and covariance matrix estimation

The problem of estimating the spectral density matrix of a multi‐variate time series is revisited with special focus on the frequencies and . Recognizing that the entries of the spectral density matrix at these two boundary points are real‐valued, we propose a new estimator constructed from a local polynomial regression of the real portion of the multi‐variate periodogram. The case is of particular importance, since is associated with the large‐sample covariance matrix of the sample mean; hence, estimating is crucial to conduct any sort of statistical inference on the mean. We explore the properties of the local polynomial estimator through theory and simulations, and discuss an application to inflation and unemployment.

Bottleneck matrices of maximal outerplanar graphs with isomorphic underlying trees

In this paper, we consider the entries of the bottleneck matrices of maximal outerplanar graphs with isomorphic underlying trees. We show how the entries of the bottleneck matrix are perturbed when we modify a maximal outerplanar graph into a nonisomorphic maximal outerplanar graph whose underlying tree is isomorphic to the underlying tree of the original maximal outerplanar graph. We illustrate our results with examples.

Value-Positivity for Matrix Games

Matrix games are the most basic model in game theory, and yet robustness with respect to small perturbations of the matrix entries is not fully understood. In this paper, we introduce value positivity and uniform value positivity, two properties that refine the notion of optimality in the context of polynomially perturbed matrix games. The first concept captures how the value depends on the perturbation parameter, and the second consists of the existence of a fixed strategy that guarantees the value of the unperturbed matrix game for every sufficiently small positive parameter. We provide polynomial-time algorithms to check whether a polynomially perturbed matrix game satisfies these properties. We further provide the functional form for a parameterized optimal strategy and the value function. Finally, we translate our results to linear programming and stochastic games, where value positivity is related to the existence of robust solutions. Funding: This research was supported by Fondation CFM pour la Recherche, the H2020 European Research Council [Grant ERC-CoG-863818 (ForM-SMArt)], the Austrian Science Fund [Grant 10.55776/COE12], ANID Chile [Grant ACT210005], and Agence Nationale de la Recherche [Grant ANR-21-CE40-0020].

Computing properties of subdivision schemes using small real Fourier indexed matrices

The quality of a subdivision scheme in the vicinity of a vertex or a face-centre is related to the eigenstructure of the subdivision matrix. When the scheme has the appropriate symmetries, a common technique, based on discrete Fourier transform, builds small complex matrices that ease the numerical analysis of the eigenelements using in particular their Fourier index. But the numerical analysis of the eigenelements remains difficult when matrix entries involve complex numbers and unknowns, for example, in cases where we are tuning a scheme. We present techniques to build similar small matrices, still associated with a Fourier index and whose eigenstructure is simply related to the full matrix, but which are real. They extend the known techniques to schemes which rotate the lattice and with vertices which do not lie topologically on symmetry axes of the studied vicinity of vertex or face centre. Our techniques make it easier to tune these subdivision schemes. We illustrate it with the analysis of the so-called Simplest Scheme at the centre of an n-sided face.

Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.

Guaranteed matrix recovery using weighted nuclear norm plus weighted total variation minimization

This work presents a general framework regarding the recovery of matrices equipped with hybrid low-rank and local-smooth properties from just a few measurements consisting of linear combinations of the matrix entries. Concretely, we consider the problem of robust low-rank matrix recovery using Weighted Nuclear Norm plus Weight Total Variation (WNNWTV) minimization. First of all, based on a new restricted isometry property, we prove that the WNNWTV method possesses an error bound consisting of a low-rank approximation term, a total variation approximation term, and an observation error term. It should be noted that although there are many models considering both properties, there are very few recoverable error theories about such models. Specifically, the theoretical error bound provides an automatic mechanism to reducing regularization parameters with no need for cross-validation while keeping almost the same selection result with commonly used cross-validation technique. Subsequently, the proposed method is reformulated into a regularized unconstrained problem, and we study its optimization aspects in detail based on the Alternating Direction Method of Multipliers (ADMM). Extensive experiments on synthetic data and two applications, i.e. hyperspectral image recovery and dynamic magnetic resonance imaging recovery verified our theories and proposed algorithms.

The impact of the European Union emissions trading system on carbon dioxide emissions: a matrix completion analysis

Despite the negative externalities on the environment and human health, today’s economies still produce excessive carbon dioxide emissions. As a result, governments are trying to shift production and consumption to more sustainable models that reduce the environmental impact of carbon dioxide emissions. The European Union, in particular, has implemented an innovative policy to reduce carbon dioxide emissions by creating a market for emission rights, the emissions trading system. The objective of this paper is to perform a counterfactual analysis to measure the impact of the emissions trading system on the reduction of carbon dioxide emissions. For this purpose, a recently-developed statistical machine learning method called matrix completion with fixed effects estimation is used and compared to traditional econometric techniques. We apply matrix completion with fixed effects estimation to the prediction of missing counterfactual entries of a carbon dioxide emissions matrix whose elements (indexed row-wise by country and column-wise by year) represent emissions without the emissions trading system for country-year pairs. The results obtained, confirmed by robust diagnostic tests, show a significant effect of the emissions trading system on the reduction of carbon dioxide emissions: the majority of European Union countries included in our analysis reduced their total carbon dioxide emissions (associated with selected industries) by about 15.4% during the emissions trading system treatment period 2005–2020, compared to the total carbon dioxide emissions (associated with the same industries) that would have been achieved in the absence of the emissions trading system policy. Finally, several managerial/practical implications of the study are discussed, together with its possible extensions.

The determinant of {±1}-matrices and oriented hypergraphs

The determinants of {±1}-matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.

Independence of linear spectral statistics and the point process at the edge of Wigner matrices

Let [Formula: see text] be a Wigner matrix of dimension [Formula: see text] with eigenvalues [Formula: see text] and [Formula: see text] be an analytic function on [Formula: see text] with polynomial growth. It is known that Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] converges in distribution to a normal random variable with mean [Formula: see text] and a finite variance depending on [Formula: see text]. On the other hand, it is also known that [Formula: see text] converges in distribution to the GOE Tracy widom law. In this paper we prove that whenever the entries of the Wigner matrix are sub-Gaussian, Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that [Formula: see text] and Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] are asymptotically independent. The main ingredient of the proof is based on a recent paper by Banerjee [A new combinatorial approach for tracy–widom law of wigner matrices, preprint (2022), arXiv:2201.00300 ]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.

Regression‐based network‐flow and inner‐matrix reconstruction

AbstractNetwork or matrix reconstruction is a general problem that occurs if the row‐ and column sums of a matrix are given, and the matrix entries need to be predicted conditional on the aggregated information. In this paper, we show that the predictions obtained from the iterative proportional fitting procedure (IPFP) or equivalently maximum entropy (ME) can be obtained by restricted maximum likelihood estimation relying on augmented Lagrangian optimization. Based on this equivalence, we extend the framework of network reconstruction, conditional on row and column sums, toward regression, which allows the inclusion of exogenous covariates and bootstrap‐based uncertainty quantification. More specifically, the mean of the regression model leads to the observed row and column margins. To exemplify the approach, we provide a simulation study and investigate interbank lending data, provided by the Bank for International Settlement. This dataset provides full knowledge of the real network and is, therefore, suitable to evaluate the predictions of our approach. It is shown that the inclusion of exogenous information leads to superior predictions in terms of and errors.

Probing the quantum nature of gravity using a Bose-Einstein condensate

The effect of noise induced by gravitons has been investigated using a Bose-Einstein condensate. The general complex scalar field theory with a quadratic self-interaction term has been considered in the presence of a gravitational wave. The gravitational wave perturbation is then considered as a sum of discrete Fourier modes in the momentum space. Varying the action and making use of the principle of least action, one obtains two equations of motion corresponding to the gravitational perturbation and the time-dependent part of the pseudo-Goldstone boson. Coming to an operatorial representation and quantizing the phase space variables via appropriately introduced canonical commutation relations between the canonically conjugate variables corresponding to the graviton and bosonic part of the total system, one obtains a proper quantum gravity setup. Then we obtain the Bogoliubov coefficients from the solution of the time-dependent part of the pseudo-Goldstone boson and construct the covariance metric for the bosons initially being in a squeezed state. The entries of the covariance matrix now involves a stochastic contribution which results in an operatorial stochastic structure of the quantum Fisher information. Using the stochastic average of the Fisher information, we obtain a lower bound on the amplitude parameter of the gravitational wave. As the entire calculation is done at zero temperature, the bosonic system, by construction, will behave as a Bose-Einstein condensate. For a Bose-Einstein condensate with a single mode, we observe that the lower bound of the expectation value of the square of the uncertainty in the amplitude measurement does not become infinite when the total observational term approaches zero. It always has a finite value if the gravitons are initially in a squeezed state with high enough squeezing. In order to sum over all possible momentum modes, we next consider a noise term with a suitable Gaussian weight factor which decays over time. We then obtain the lower bound on the final expectation value of the square of the variance in the amplitude parameter. Because of the noise induced by the graviton, there is a minimum value of the measurement time below which it is impossible to detect any gravitational wave using a Bose-Einstein condensate. Finally, we consider interaction between the phonon modes of the Bose-Einstein condensate which results in a decoherence. We observe that the decoherence effect becomes significant for gravitons with minimal squeezing. Published by the American Physical Society 2024

Comparison of different cross-sectional approaches for the structural design and optimization of composite wind turbine blades based on beam models

Abstract. During the preliminary design phase of wind turbine blades, the evaluation of many design candidates in a short period of time plays an important role. Computationally efficient methods for the structural analysis that correctly predict stiffness matrix entries for beam models including the (bend–twist) coupling terms are thus needed. The present paper provides an extended overview of available approaches and shows their abilities to fulfill the requirements for the composite design of rotor blades with respect to accuracy and computational efficiency. Three cross-sectional theories are selected and implemented to compare the prediction quality of the cross-sectional coupling stiffness terms and the stress distribution based on different multi-cell test cross-sections. The cross-sectional results are compared with the 2D finite element code BECAS and are discussed in the context of accuracy and computational efficiency. The analytical solution performing best shows very small deviations in the stiffness matrix entries compared to BECAS (below 1 % in the majority of test cases). It achieved a better resolution of the stress distribution and a computation time that is more than an order of magnitude smaller using the same spatial discretization. The deviations of the stress distributions are below 10 % for most test cases. The analytical solution can thus be rated as a feasible approach for a beam-based pre-design of wind turbine rotor blades.

Generative adversarial networks for multi-fidelity matrix completion with massive missing entries

Multi-fidelity matrices refer to a pair of data matrices, whose entries are the measurements of a specific physical quantity at different fidelity levels w.r.t. two environment variables arranged as the row and the column of matrices, respectively. Although a high-fidelity (HF) matrix can provide an accurate description of the physical quantity, it usually has a number of missing entries because of high acquisition cost. In contrast, its low-fidelity (LF) counterpart is complete but the accuracy of its entries cannot meet the actual requirement. Multi-fidelity matrix completion (MMC) aims to complete the missing entries of the HF matrix by fusing the LF matrix and the available entries of the HF matrix. In this paper, we develop a generative adversarial network for multi-fidelity matrix completion (GAN-MMC), which is composed of a generator and a discriminator. The generator input is the LF matrix and its output is the approximation of the complete HF matrix. The discriminator identifies whether the HF approximation completed by the generator accords with the real HF data distribution. A matrix data augmentation method is proposed to enrich the samples for training GAN-MMC. Moreover, a correlation coefficient-based term is also imposed into the generative loss in order to make the generator capture the relatedness information between the available HF entries and the corresponding LF entries. The numerical experiments show that the proposed model can effectively complete the HF matrix with massive missing entries.

Differentiable microstructures design via anisotropic thermal diffusion

We propose a novel method to design differentiable microstructures. Central to our algorithm is a new representation of the mapping from the parameters to microstructures, formulated as the anisotropic thermal diffusion. A metric field governs the anisotropic diffusion. The metric associated with each point is represented as a 2 × 2 symmetric positive definite matrix that becomes the design variable. To alleviate the difficulties caused by symmetric positive definite constraints, we perform the singular value decomposition of the metric matrix so that the design variable includes a rotation angle and a diagonal matrix. Then, the positive definiteness is converted to requiring the two diagonal entries of the diagonal matrix to be positive, which is easier to deal with. The effectiveness of our algorithm is demonstrated through evaluations and comparisons over various examples.

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

In this paper, we derive novel families of inclusion sets for the spectra and pseudospectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687–704].

On the capacity of artificial intelligence techniques and statistical methods to deal with low-quality data in medical supply chain environments

We illustrate the capacity of Artificial Intelligence (AI) and Machine Learning (ML) techniques to preserve consistent categorization abilities whenever the quality of the data decreases, displaying mistakes or mismatches across matrix entries, while standard statistical methods exhibit significant modifications in the value of the corresponding coefficients. We design algorithms of different complexity to generate a series of comparable profiles. These profiles are compared within environments that allow for an immediate identification of the generating algorithms and within increasingly complex settings involving almost identical profiles derived from different algorithms. AI and ML techniques outperform standard statistical methods when distinguishing the algorithms generating the profiles. Building on these results, we perform a retrospective analysis where AI and ML techniques are applied to two empirical scenarios defined by different data series of patients transplanted through the period 2006–2019. The first scenario contains the variables describing the evolution of patients inputted correctly. In the second, we modify the content of the vectors of characteristics defining the evolution of patients by exchanging the values of a subset of realizations from two categorical variables. AI and ML techniques are consistently accurate when categorizing patients correctly within both scenarios, a feature particularly relevant when the quality of the information sources composing the medical chain varies. This latter problem is exacerbated among hospitals located in developing countries, where the quality of the data gathered limits their identification and extrapolation capacities.

Regionalization-Based Collaborative Filtering: Harnessing Geographical Information in Recommenders

Regionalization, also known as spatially constrained clustering , is an unsupervised machine learning technique used to identify and define spatially contiguous regions. In this work, we introduce a methodology to regionalize recommendation systems (RSs) based on a collaborative filtering approach. Two main challenges arise when performing regionalization based on users’ preferences in RSs: (1) unstructured data, as interactions are often scarce and observed on a smaller scale; and (2) the difficulty of evaluation of the quality of the clustering results. To address these challenges, our methodology relies on inductive matrix completion (IMC), a fundamental approach to recover unknown entries of a rating matrix while utilizing region information to extract a region-based feature matrix. With this feature matrix, our method becomes adaptive and seamlessly integrates with various regionalization algorithms to create regionalization candidates. This enables us to derive more accurate recommendations that consider regionalized effects and discover interesting patterns in localized user behavior. We experimentally evaluate our model on synthetic datasets to demonstrate its efficacy in settings where our underlying assumptions are correct. Furthermore, we present a real-world case study illustrating the interpretable information the model can derive in terms of regionalized recommendation relevance.

Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians

Based oh the properties of Lie algebras, in this work we develop a general framework to linearize the von Neumann equation rendering it in a suitable form for quantum simulations. Departing from the conventional method of expanding the density matrix in the Liouville space formed by matrices unit we express the von Neumann equation in terms of Pauli strings. This provides several advantages related to the quantum tomography of the density matrix and the formulation of the unitary gates that generate the time evolution. The use of Pauli strings facilitates the quantum tomography of the density matrix whose elements are purely real. As for any other basis of Hermitian matrices, this eliminates the need to calculate the phase of the complex entries of the density matrix. This approach also enables to express the evolution operator as a sequence of commuting Hamiltonian gates of Pauli strings that can readily be synthetized using Clifford gates. Additionally, the fact that these gates commute with each other along with the unique properties of the algebra formed by Pauli strings allows to avoid the use of Trotterization hence considerably reducing the circuit depth. The algorithm is demonstrated for three Hamiltonians using the IBM noisy quantum circuit simulator.

On the two-loop penguin contributions to the Anomalous Dimensions of four-quark operators

We revisit the Next-to-Leading Order (two-loop) contributions to the Anomalous Dimensions of ∆F = 1 four-quark operators in QCD. We devise a test for anomalous dimensions, that we regard as of general interest, and by means of which we detect a problem in the results available in the literature. Deconstructing the steps leading to the available result, we identify the source of the problem, which is related to the operator known as Q11. We show how to fix the problem and provide the corrected anomalous dimensions. With the insight of our findings, we propose an alternative approach to the one used in the literature which does not suffer from the identified disease, and which confirms our corrected results. We assess the numerical impact of our corrections, which happens to be in the ballpark of 5% in certain entries of the evolution matrix. Our results are important for the correct resummation of Next-to-Leading Logarithms in analyses of physics beyond the Standard Model in ∆F = 1 processes, such as the decays of Kaons and B-mesons.