Querying a Matrix through Matrix-Vector Products

We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

Spectral properties for the Laplacian of a generalized Wigner matrix

In this paper, we consider the spectrum of a Laplacian matrix, also known as Markov matrices where the entries of the matrix are independent but have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that the variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution converges. We give an expression for the moments of the limiting measure in terms of graph homomorphisms. In some special cases, we identify the limit explicitly. We also study the spectral norm and derive the order of the maximum eigenvalue. We show that our results cover Laplacians of various random graphs including inhomogeneous Erdős–Rényi random graphs, sparse W-random graphs, stochastic block matrices and constrained random graphs.

Synchronizability analysis of three kinds of dynamical weighted fractal networks

In this paper, we study the synchronizability of three kinds of dynamical weighted fractal networks (WFNs). These WFNs are weighted Cantor-dust networks, weighted Sierpinski networks and weighted Koch networks. We calculated some features of these WFNs, including average distance ([Formula: see text]), fractal dimension ([Formula: see text]), information dimension ([Formula: see text]), correlation dimension ([Formula: see text]). We analyze two representative types of synchronizable dynamical networks (the type-I and the type-II). There are two indexes ([Formula: see text] and [Formula: see text]) that can be used to characterize the synchronizability of the two types of dynamical network. Here, [Formula: see text] and [Formula: see text] are the minimum nonzero eigenvalue and the maximum eigenvalue of the Laplacian matrix of the network, respectively. We find that the larger scaling factor [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] implies stronger synchronizability for the type-I dynamical WFNs.

Constrained Eigenvalue Minimization of Incomplete Pairwise Comparison Matrices by Nelder-Mead Algorithm

Pairwise comparison matrices play a prominent role in multiple-criteria decision-making, particularly in the analytic hierarchy process (AHP). Another form of preference modeling, called an incomplete pairwise comparison matrix, is considered when one or more elements are missing. In this paper, an algorithm is proposed for the optimal completion of an incomplete matrix. Our intention is to numerically minimize a maximum eigenvalue function, which is difficult to write explicitly in terms of variables, subject to interval constraints. Numerical simulations are carried out in order to examine the performance of the algorithm. The results of our simulations show that the proposed algorithm has the ability to solve the minimization of the constrained eigenvalue problem. We provided illustrative examples to show the simplex procedures obtained by the proposed algorithm, and how well it fills in the given incomplete matrices.

THE TARGET DETECTION ON THE SEA SURFACE BASED ON STANDARD DEVIATION OF THE MAXIMUM EIGENVALUE OF THE POLARIZATION COVARIANCE MATRIX

This paper proposes a novel method to detect small sea targets, using polarimetric radar. The standard deviation of maximum eigenvalue of the polarization covariance matrix is ultilized for the detection. In order to evaluate the new algorithm in practical applications, the real data collected from polarimetric IPIX radar at MC Master University is tested with the new algorithm, and the result is displayed on the radar screen. The results of this method is also compared with those of the DoP and SP-GLR methods. Initial results show that the novel method significantly improves the detectability of small sea targets on the sea surface.

An Integrated Parallel Multistage Spectrum Sensing for Cognitive Radio

Spectrum sensing for cognitive radio requires speed and good detection performance at very low SNR ratios. There is no single-stage spectrum sensing technique that is perfect enough to be implemented in practical cognitive radio. In this paper, the authors propose a new parallel fully blind multistage detector. They assume the appropriate stage based on the estimated SNR values that are achieved from the SNR estimator. Energy detection is used in first stage for its simplicity and sensing accuracy at high SNR. For low SNRs, they adopt the maximum eigenvalues detector with different smoothing factor in higher stages. The sensing accuracy for the maximum eigenvalue detector technique improves with higher value of the smoothing factor. However, the computational complexity will increase significantly. They analyze the performance of two cases of the proposed detector: two-stage and three-stage schemes. The simulation results show that the proposed detector improves spectrum sensing in terms of accuracy and speed.

Analysis of grid operation state based on improved maximum eigenvalue sample covariance matrix algorithm

Traditional static threshold–based state analysis methods can be applied to specific signal-to-noise ratio situations but may present poor performance in the presence of large sizes and complexity of power system. In this article, an improved maximum eigenvalue sample covariance matrix algorithm is proposed, where a Marchenko–Pastur law–based dynamic threshold is introduced by taking all the eigenvalues exceeding the supremum into account for different signal-to-noise ratio situations, to improve the calculation efficiency and widen the application fields of existing methods. The comparison analysis based on IEEE 39-Bus system shows that the proposed algorithm outperforms the existing solutions in terms of calculation speed, anti-interference ability, and universality to different signal-to-noise ratio situations.

Multi-asset generalized variance swaps in Barndorff-Nielsen and Shephard model

This paper proposes swaps on two important new measures of generalized variance, namely, the maximum eigenvalue and trace of the covariance matrix of the assets involved. We price these generalized variance swaps for Barndorff-Nielsen and Shephard model used in financial markets. We consider multiple assets in the portfolio for theoretical purpose and demonstrate our approach with numerical examples taking three stocks in the portfolio. The results obtained in this paper have important implications for the commodity sector where such swaps would be useful for hedging risk.