Eigenvector Structure Research Articles

Chaos in the three-site Bose-Hubbard model: Classical versus quantum.

We consider a quantum many-body system-the Bose-Hubbard system on three sites-which has a classical limit, and which is neither strongly chaotic nor integrable but rather shows a mixture of the two types of behavior. We compare quantum measures of chaos (eigenvalue statistics and eigenvector structure) in the quantum system, with classical measures of chaos (Lyapunov exponents) in the corresponding classical system. As a function of energy and interaction strength, we demonstrate a strong overall correspondence between the two cases. In contrast to both strongly chaotic and integrable systems, the largest Lyapunov exponent is shown to be a multivalued function of energy.

Robust Precoding for 3D Massive MIMO Configuration With Matrix Manifold Optimization

This paper investigates robust downlink precoding for three-dimensional (3D) massive multi-input multi-output (MIMO) configuration with matrix manifold optimization. Starting with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> channel model, we formulate the robust precoder design to maximize an upper bound of ergodic weighted sum-rate under a total power budget. We derive the generalized eigenvector structure for optimal precoder with matrix manifold optimization. However, since the precoding of multiple users is coupled in the structure, we maximize the objective function for each user in alternation and prove the solution of each individual problem is the generalized eigenvector corresponding to the maximum generalized eigenvalue. In accordance with this, we design an iterative algorithm and present its convergence analysis. Furthermore, we propose a Riemannian conjugate gradient (RCG) method to solve the generalized eigenvalue problem (GEP) for higher efficiency in the precoder design algorithm.

Edge-connectivity matrices and their spectra

The edge-connectivity matrix of a weighted graph is the matrix whose off-diagonal v-w entry is the weight of a minimum edge cut separating vertices v and w. Its computation is a classical topic of combinatorial optimization since at least the seminal work of Gomory and Hu. In this article, we investigate spectral properties of these matrices. In particular, we provide tight bounds on the smallest eigenvalue and the energy. Moreover, we study the eigenvector structure and show in which cases eigenvectors can be easily obtained from matrix entries. These results in turn rely on a new characterization of those nonnegative matrices that can actually occur as edge-connectivity matrices.

A comparison of approximate non-linear Riemann solvers for Relativistic MHD

ABSTRACT We compare a particular selection of approximate solutions of the Riemann problem in the context of ideal relativistic magnetohydrodynamics. In particular, we focus on Riemann solvers not requiring a full eigenvector structure. Such solvers recover the solution of the Riemann problem by solving a simplified or reduced set of jump conditions, whose level of complexity depends on the intermediate modes that are included. Five different approaches – namely the HLL, HLLC, HLLD, HLLEM, and GFORCE schemes – are compared in terms of accuracy and robustness against one – and multidimensional standard numerical benchmarks. Our results demonstrate that – for weak or moderate magnetizations – the HLLD Riemann solver yields the most accurate results, followed by HLLC solver(s). The GFORCE approach provides a valid alternative to the HLL solver being less dissipative and equally robust for strongly magnetized environments. Finally, our tests show that the HLLEM Riemann solver is not cost-effective in improving the accuracy of the solution and reducing the numerical dissipation.

Identifying vital nodes from local and global perspectives in complex networks

Recognition of vital nodes in complex networks retains great importance in the improvement of network’s robustness and vulnerability. Consistent research proposed various approaches like local-structure-based methods, e.g., degree centrality, pagerank, etc., and global-structure-based methods, e.g., betweenness, closeness centrality, etc., to evaluate the concerned nodes. Though their performance is amazingly well, these methods have undergone some intrinsic limitations. For instance, local-structure-based methods lose some sort of global information and global-structure-based methods are too complicated to measure the important nodes, particularly in networks where sizes become large. To tackle these challenges, we propose a Local-and-Global-Centrality (LGC) measuring algorithm to identify the vital nodes through handling local as well as global topological aspects of a network simultaneously. In order to assess the performance of the proposed algorithm with respect to the state-of-the-art methodologies, we performed experiments through LCG, Betweenness (BNC), Closeness (CNC), Gravity (GIC), Page-Rank (PRC), Eigenvector (EVC), Global and Local Structure (GLS), Global Structure Model (GSM), and Profit-leader (PLC) methods on differently sized real-world networks. Our experiments disclose that LGC outperformed many of the compared techniques.

On eigenvector structure of weakly balanced networks

A signed network is called weakly balanced when its vertex set can be partitioned into two or more vertex subsets (clusters) such that every positive edge joins the vertices of the same subset and every negative edge joins the vertices of different subsets. Finding a spectral criterion for weakly balanced networks is an open problem. In this paper, the eigenvector structure of some weakly balanced signed networks is calculated. The eigenvector structure helps in the transformation of (−1,0,1)-adjacency matrices of these networks to tridiagonal matrices. Hence, we provide partial results on eigenvalues of these networks, and in general, their spectra can be calculated using the existing numerical methods.

Spatial search using flip-flop quantum walk

We analyse the eigenvalue and eigenvector structure of the flip-flop quantum walk on regular graphs, explicitly demonstrating how it is quadratically faster than the classical random walk. Then we use it in a controlled spatial search algorithm with multiple target states, and determine the oracle complexity as a function of the spectral gap and the number of target states. The oracle complexity is optimal as a function of the graph size and the number of target states, when the spectral gap of the adjacency matrix is $\Theta(1)$. It is also optimal for spatial search on D&gt;4 dimensional hypercubic lattices. Otherwise it matches the best result available in the literature, with a much simpler algorithm. Our results also yield bounds on the classical hitting time of random walks on regular graphs, which may be of independent interest.

Clustering Based on Eigenvectors of the Adjacency Matrix

Abstract The paper presents a novel spectral algorithm EVSA (eigenvector structure analysis), which uses eigenvalues and eigenvectors of the adjacency matrix in order to discover clusters. Based on matrix perturbation theory and properties of graph spectra we show that the adjacency matrix can be more suitable for partitioning than other Laplacian matrices. The main problem concerning the use of the adjacency matrix is the selection of the appropriate eigenvectors. We thus propose an approach based on analysis of the adjacency matrix spectrum and eigenvector pairwise correlations. Formulated rules and heuristics allow choosing the right eigenvectors representing clusters, i.e., automatically establishing the number of groups. The algorithm requires only one parameter-the number of nearest neighbors. Unlike many other spectral methods, our solution does not need an additional clustering algorithm for final partitioning. We evaluate the proposed approach using real-world datasets of different sizes. Its performance is competitive to other both standard and new solutions, which require the number of clusters to be given as an input parameter.

Are the eigenvalues of the B‐spline isogeometric analysis approximation of −Δu = λu known in almost closed form?

SummaryWe consider the B‐spline isogeometric analysis approximation of the Laplacian eigenvalue problem −Δu = λu over the d‐dimensional hypercube (0,1)d. By using tensor‐product arguments, we show that the eigenvalue–eigenvector structure of the resulting discretization matrix is completely determined by the eigenvalue–eigenvector structure of the matrix arising from the isogeometric analysis approximation based on B‐splines of degree p of the unidimensional problem . Here, n is the mesh fineness parameter, and the size of is N(n,p) = n + p − 2. In previous works, it was established that the normalized sequence enjoys an asymptotic spectral distribution described by a function ep(θ), the so‐called spectral symbol. The contributions of this paper can be summarized as follows: We prove several important analytic properties of the spectral symbol ep(θ). In particular, we show that ep(θ) is monotone increasing on [0,π] for all p ≥ 1 and that ep(θ)→θ2 uniformly on [0,π] as p→∞. For p = 1 and p = 2, we show that belongs to a matrix algebra associated with a fast unitary sine transform, and we compute eigenvalues and eigenvectors of . In both cases, the eigenvalues are given by ep(θj,n), j = 1,…,n + p − 2, where θj,n = jπ/n. For p ≥ 3, we provide numerical evidence of a precise asymptotic expansion for the eigenvalues of , excluding the largest eigenvalues (the so‐called outliers). More precisely, we numerically show that for every p ≥ 3, every integer α ≥ 0, every n, and every , urn:x-wiley:nla:media:nla2198:nla2198-math-0010 where the eigenvalues of are arranged in ascending order, ; is a sequence of functions from [0,π] to , which depends only on p; h = 1/n and θj,n = jπh for j = 1,…,n; and is the remainder, which satisfies for some constant depending only on α and p. We also provide a proof of this expansion for α = 0 and j = 1,…,N(n,p) −(4p − 2), where 4p − 2 represents a theoretical estimate of the number of outliers . We show through numerical experiments that, for p ≥ 3 and k ≥ 1, there exists a point θ( p,k) ∈ (0,π) such that vanishes on [0,θ( p,k)]. Moreover, as it is suggested by the numerics of this paper, the infimum of θ(p,k) over all k ≥ 1, say yp, is strictly positive, and the equation holds numerically whenever θj,n &lt; θ( p), where θ( p) is a point in (0,yp] which grows with p. For p ≥ 3, based on the asymptotic expansion in the above item 3, we propose a parallel interpolation–extrapolation algorithm for computing the eigenvalues of , excluding the outliers. The performance of the algorithm is illustrated through numerical experiments. Note that, by the previous item 4, the algorithm is actually not necessary for computing the eigenvalues corresponding to points θj,n &lt; θ( p).

Essential core of the Hawking–Ellis types

The Hawking–Ellis (Segre–Plebański) classification of possible stress–energy tensors is an essential tool in analyzing the implications of the Einstein field equations in a more-or-less model-independent manner. In the current article the basic idea is to simplify the Hawking–Ellis type I, II, III, and IV classification by isolating the ‘essential core’ of the type II, type III, and type IV stress–energy tensors; this being done by subtracting (special cases of) type I to simplify the (Lorentz invariant) eigenvalue structure as much as possible without disturbing the eigenvector structure. We will denote these ‘simplified cores’ type II0, type III0, and type IV0. These ‘simplified cores’ have very nice and simple algebraic properties. Furthermore, types I and II0 have very simple classical interpretations, while type IV0 is known to arise semi-classically (in renormalized expectation values of standard stress–energy tensors). In contrast type III0 stands out in that it has neither a simple classical interpretation, nor even a simple semi-classical interpretation. We will also consider the robustness of this classification considering the stability of the different Hawking–Ellis types under perturbations. We argue that types II and III are definitively unstable, whereas types I and IV are stable.

Mode decoupling robust eigenstructure assignment applied to the lateral-directional dynamics of the F-16 aircraft

In this paper, a robust controller is designed which is based on the eigenstructure assignment technique. With this technique, the physical modes of the system are decoupled by the appropriate shaping of the eigenvectors. While keeping the desired eigenvector structure as much as possible, Nelder–Mead optimization algorithm is implemented for the search of the optimal robust solution. Different optimization cost functions are investigated in terms of eigenvalue sensitivity and stability robustness. Design solutions are compared with each other and with a fundamental solution in terms of robustness measures. Furthermore, a small amount of flexibility is included in the selection of the eigenvalue locations and the improvement in the robustness is evaluated by singular value analysis. The effectiveness of the control system is demonstrated with nonlinear simulations on the F-16 aircraft mathematical model.

Regularized covariance matrix estimation under the common principal components model

ABSTRACTThe common principal components (CPC) model provides a way to model the population covariance matrices of several groups by assuming a common eigenvector structure. When appropriate, this model can provide covariance matrix estimators of which the elements have smaller standard errors than when using either the pooled covariance matrix or the per group unbiased sample covariance matrix estimators. In this article, a regularized CPC estimator under the assumption of a common (or partially common) eigenvector structure in the populations is proposed. After estimation of the common eigenvectors using the Flury–Gautschi (or other) algorithm, the off-diagonal elements of the nearly diagonalized covariance matrices are shrunk towards zero and multiplied with the orthogonal common eigenvector matrix to obtain the regularized CPC covariance matrix estimates. The optimal shrinkage intensity per group can be estimated using cross-validation. The efficiency of these estimators compared to the pooled and unbiased estimators is investigated in a Monte Carlo simulation study, and the regularized CPC estimator is applied to a real dataset to demonstrate the utility of the method.

Extending Classical Multirate Signal Processing Theory to Graphs—Part II: M-Channel Filter Banks

This paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper (Part I) and studies $M$ -channel polynomial filter banks on graphs. The behavior of such graph filter banks differs from that of classical filter banks in many ways, the precise details depending on the eigenstructure of the adjacency matrix $\boldsymbol {A}$ . It is shown that graph filter banks represent, (linear and) periodically, shift-variant systems only when $\boldsymbol {A}$ satisfies the noble identity conditions developed in Part I. It is then shown that perfect reconstruction graph filter banks can always be developed when $\boldsymbol {A}$ satisfies the eigenvector structure satisfied by $M$ -block cyclic graphs and has distinct eigenvalues (further restrictions on eigenvalues being unnecessary for this). If $\boldsymbol {A}$ is actually $M$ -block cyclic then these PR filter banks indeed become practical, i.e., arbitrary filter polynomial orders are possible, and there are robustness advantages. In this case, the PR condition is identical to PR in classical filter banks—any classical PR example can be converted to a graph PR filter bank on an $M$ -block cyclic graph. It is shown that for $M$ -block cyclic graphs with all eigenvalues on the unit circle, the frequency responses of filters have meaningful correspondence with classical filter banks. Polyphase representations are then developed for graph filter banks and utilized to develop alternate conditions for alias cancellation and perfect reconstruction, again for graphs with specific eigenstructures. It is then shown that the eigenvector condition on the graph can be relaxed by using similarity transforms.

Graph Edit Distance: Moving from global to local structure to solve the graph-matching problem

Classical graph approaches for pattern recognition applications rely on computing distances between graphs in a certain graph domain. That is, the distance between two graphs is obtained by directly optimising some objective function, which consider node and edge attributes. Bipartite Graph Matching was first published in a journal in 2009 and new versions have appeared to speed up its runtime such as the Fast Bipartite Graph Matching. This algorithm is based on defining a cost matrix between the whole nodes of both graphs and solving the node correspondence through a linear assignment method. To construct the matrix, several local structures can be defined from the simplest one (only the node) to the most complex (a whole clique or eigenvector structure). In this paper, we propose eight different options and we show that the type of local structure and the distance defined between these structures is relevant for the runtime and classification ratio.

Robust eigenstructure clustering by non-smooth optimisation

We extend classical eigenstructure assignment to more realistic problems, where additional performance and robustness specifications arise. Our aim is to combine time-domain constraints, as reflected by pole location and eigenvector structure, with frequency-domain objectives such as the H2, H∞ or Hankel norms. Using pole clustering, we allow poles to move in polydisks of prescribed size around their nominal values, driven by optimisation. Eigenelements, that is poles and eigenvectors, are allowed to move simultaneously and serve as decision variables in a specialised non-smooth optimisation technique. Two aerospace applications illustrate the power of the new method.

Spectral backbone of excitation transport in ultracold Rydberg gases

The spectral structure underlying excitonic energy transfer in ultracold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics, in particular, the density of states and the nearest-neighbor level-spacing statistics, exhibits a transition from approximately a 1-stable L\'evy to a Gaussian orthogonal ensemble. Deviations from random matrix statistics are shown to stem from correlations between interatomic interaction strengths that lead to an asymmetry of the spectral density and profoundly affect localization properties. We discuss approximations to the self-consistent Matsubara-Toyozawa locator expansion that incorporate these effects.

Optimization of Input Covariance Matrix for Multi-Antenna Correlated Channels

Optimizing the input covariance matrix of a multiple-antenna transmit system with partial channelstructure feedback is an important issue to fully exploit the channel capacity. Efficient design of the optimal input covariance matrix, however, remains unavailable although its eigenvector structure was clearly revealed in a recent publication. In this paper, we obtain an explicit derivative function forming a solid basis for optimizing the optimal input covariance matrix. This new derivative expression enables us to further develop an efficient iterative algorithm for determining the optimal eigenvalues. The technique is illustrated through numerical examples.

Discriminating the Weyl type in higher dimensions using scalar curvature invariants

We consider higher dimensional Lorentzian spacetimes which are currently of interest in theoretical physics. It is possible to algebraically classify any tensor in a Lorentzian spacetime of arbitrary dimensions using alignment theory. In the case of the Weyl tensor, and using bivector theory, the associated Weyl curvature operator will have a restricted eigenvector structure for algebraic types II and D, which leads to necessary conditions on the discriminants of the associated characteristic equation which can be manifestly expressed in terms of polynomial scalar curvature invariants. The use of such necessary conditions in terms of scalar curvature invariants will be of great utility in the study and classification of black hole solutions in more than four dimensions.

Inspection of hexagonal and triangular C-grid discretizations of the shallow water equations

C-grid discretizations based on a hexagonal or triangular mesh can be investigated with the help of a planar trivariate coordinate system, where the vector components are either defined tangentially (hexagonal C-grid) or perpendicularly (triangular C-grid) to the coordinate lines. Inspecting the Helmholtz decomposition of a vector in case of the linearly dependent trivariate coordinate description reveals insights into the structure and stencil of the discretized divergence and vorticity on such grids. From a vector Laplacian, which is consistent with the Helmholtz decomposition, a general formulation for the inner product operator at grid edges can be derived. Thus, the vector reconstruction of the tangential wind for the Coriolis term in the shallow water equations can be given even for a slightly distorted tesselation as present on icosahedral spherical grids. Furthermore, a rigorous comparison of the triangular and the hexagonal C-grid linear shallow water equations is performed from a theoretical and also from an experimental viewpoint. It turns out that the additional degree of freedom in the height field in the triangular C-grid case compared to the hexagonal C-grid is responsible for the decoupling of divergence values on upward and downward directed triangles. This problem occurs especially for small Rossby deformation radii, and practically requires additional explicit diffusion. In contrast, the hexagonal C-grid discretization has remarkable similarity to the quadrilateral C-grid case in the wave dispersion properties and eigenvector structure. Numerical experiments performed with that option proved resilent.

Effect on surface ship interference on adaptive beamforming for the shallow water array performance (SWAP) project

The shallow water array performance (SWAP) project is designed to explore the limits of large-aperture passive sonar array processing capability in a shallow-water environment with moving surface ship interference. The array, off the eastern coast of Florida near Ft. Lauderdale, has 500 elements with a total length of approximately 900 m. One challenge for the processing is the length of observation time needed by traditional adaptive beamforming formulations—with an array of this size and element quantity, the resolution of range and bearing cells is such that a ship may move across many cells during the snapshot time, spreading the resulting eigenvector structure and decreasing effective signal gain. A central question is the trade-off between array gain and this eigenvector spreading loss. To explore this question, tracks from actual ships in the vicinity of the array site, combined with sound speed and bathymetry data from the site, are used in an adiabatic normal mode simulation to predict the acoustic response across the array. The resulting simulated snapshots are used in adaptive and non-adaptive formulations to predict target detection performance as a function of the interference environment and processing parameters, for both full and sub-aperture processing schemes.