Marcenko-Pastur Law Research Articles

Intrusion Monitoring Based on High Dimensional Random Matrix by Using Ultra-Weak Fiber Bragg Grating Array

In order to ensure that a perimeter security system can work effectively, a convenient and effective event detection algorithm has an important engineering significance. Given the above background, in this paper, we propose a high reliability intrusion event recognition method and vibration sensing system, based on ultra-weak fiber Bragg grating array, by using high dimensional random matrix. We obtain a high sensitivity optical interference signal by constructing a patch-matched optical interference system, then compose the demodulated interference signal into a high-dimensional random matrix. The statistical characteristics of the matrix for the Marcenko-Pastur (M-P) law and ring law are used to confirm the presence of intrusion events efficiently, which can reflect the limit spectrum distribution of the high-dimensional random matrix; meanwhile, the abnormal state quantity and moment are obtained. Further, the average spectral radius value is used to judge the fault cause. Field experimental results show that the proposed method can effectively obtain the correct monitoring data for the sensor array. By comparing the monitoring results of normal operation and crusher operation, we can detect the intrusion event in 4.5 s, and the accuracy rate can reach more than 90%, which verifies that the proposed high-dimensional random matrix analysis method can work properly, proving a practical engineering application prospect.

Capacity Analysis and Optimal Spacing Design for Compact Array Massive MIMO Systems With Finite Aperture

To achieve large-scale antenna deployment in limited space, compact arrays should be widely used. In this letter, we discuss the mathematical relationship between system capacity and the aperture of compact uniform linear arrays (ULAs). In particular, the theoretical analysis benefits from the Marcenko-Pastur law and Toeplitz feature of the mutual coupling and correlation matrices. The element spacing optimization problem is then formulated to maximize the system capacity under a finite aperture. After illustrating its convexity, the optimal spacing can be obtained by applying one-dimensional search methods. Further, the discussions can be readily extended to compact uniform planar arrays (UPAs) with finite apertures. Numerical results show the superiority of optimal spacing in system capacity for both ULAs and UPAs to half-wavelength spacing.

Efficient GNSS Jamming Mitigation Using the Marcenko Pastur Law and Karhunen–Loeve Decomposition

Efficient GNSS Jamming Mitigation Using the Marcenko Pastur Law and Karhunen–Loeve Decomposition

The PPT square conjecture holds generically for some classes of independent states

Let be a random pure state on , where ψ is a random unit vector uniformly distributed on the sphere in . Let be a random induced state whose distribution is , and let be a random induced state following the same distribution independent of . Let ρ be a random state induced by the entanglement swapping of and . We show that the empirical spectrum of converges almost surely to the Marcenko–Pastur law with parameter c2 as and . As an application, we prove that the state ρ is separable generically if are PPT entangled.

Spectral analysis of high-dimensional sample covariance matrices with missing observations

We study high-dimensional sample covariance matrices based on independent random vectors with missing coordinates. The presence of missing observations is common in modern applications such as climate studies or gene expression micro-arrays. A weak approximation on the spectral distribution in the “large dimension $d$ and large sample size $n$” asymptotics is derived for possibly different observation probabilities in the coordinates. The spectral distribution turns out to be strongly influenced by the missingness mechanism. In the null case under the missing at random scenario where each component is observed with the same probability $p$, the limiting spectral distribution is a Marcenko–Pastur law shifted by $(1-p)/p$ to the left. As $d/n\rightarrow y\in(0,1)$, the almost sure convergence of the extremal eigenvalues to the respective boundary points of the support of the limiting spectral distribution is proved, which are explicitly given in terms of $y$ and $p$. Eventually, the sample covariance matrix is positive definite if $p$ is larger than \[1-(1-\sqrt{y})^{2},\] whereas this is not true any longer if $p$ is smaller than this quantity.

A Random Matrix Approach to the Forensic Analysis of Upscaled Images

The forensic analysis of resampling traces in upscaled images is addressed via subspace decomposition and random matrix theory principles. In this context, we derive the asymptotic eigenvalue distribution of sample autocorrelation matrices corresponding to genuine and upscaled images. To achieve this, we model genuine images as an autoregressive random field and we characterize upscaled images as a noisy version of a lower dimensional signal. Following the intuition behind Marcenko-Pastur law, we show that for upscaled images, the gap between the eigenvalues corresponding to the low-dimensional signal and the ones from the background noise can be enhanced by extracting a small number of consecutive columns/rows from the matrix of observations. In addition, using bounds provided by the same law for the eigenvalues of the noise space, we propose a detector for exposing traces of resampling. Finally, since an interval of plausible resampling factors can be inferred from the position of the gap, we empirically demonstrate that by using the resulting range as the search space of existing estimators (based on different principles), a better estimation accuracy can be attained with respect to the standalone versions of the latter.

Recent developments of constructing adjacency matrix in network analysis

Abstract In this paper, we review recent developments in network analysis using the graphtheory, and introduce ongoing research area with relevant theoretical results. In spe-ci c, we introduce basic notations in graph, and conditional and marginal approach inconstructing the adjacency matrix. Also, we introduce the Marcenko-Pastur law, theTracy-Widom law, the white Wishart distribution, and the spiked distribution. Finally,we mention the relationship between degrees and eigenvalues for the detection of hubsin a network.Keywords: Adjacency matrix, conditional dependency, graph theory, marginal depen-dency, Tracy-Widom law. 1. Introduction Recently Big data is one of the most hot issues in many branches of science, and complexnetwork problem is the central issue in Big data. Lots of interest arose in complex networkssuch as the world-wide web, the internet, biological networks, social networks, and so on. Awonderful scienti c fact is that any complex network can be represented by a graph, and anygraph can be represented by an adjacency matrix. Further, degree matrix and/or Laplacianmatrix are derived by the adjacency matrix.Graph theory has been developed by mathematicians for a long time, and mathematiciansare interested in properties of eigenvalues of adjacency matrix, distribution of eigenvalues(Wigner, 1955; Marcenko and Pastur, 1967) and distribution of the largest eigenvalue (Tracyand Widom, 1996), lower and/or upper bound of eigenvalues, relationship between degreeand eigenvalues, and so on. There are numerous books on the graph theory, and one of therecent and best references is Mieghem (2010).On the other hand, statisticians paid attention to the graph theory quite recently, and theyare mainly interested in graphical models and estimation of adjacency matrix using availableobservations. Recently, statisticians are interested in the case where p(number of variables)is larger than n(sample size) because lots of recent networks such as biological networks and

Universality of covariance matrices

In this paper we prove the universality of covariance matrices of the form $H_{N\times N}={X}^{\dagger}X$ where $X$ is an ${M\times N}$ rectangular matrix with independent real valued entries $x_{ij}$ satisfying $\mathbb{E}x_{ij}=0$ and $\mathbb{E}x^{2}_{ij}={\frac{1}{M}}$, $N,M\to \infty$. Furthermore it is assumed that these entries have sub-exponential tails or sufficiently high number of moments. We will study the asymptotics in the regime $N/M=d_{N}\in(0,\infty),\lim_{N\to\infty}d_{N}\neq0,\infty$. Our main result is the edge universality of the sample covariance matrix at both edges of the spectrum. In the case $\lim_{N\to\infty}d_{N}=1$, we only focus on the largest eigenvalue. Our proof is based on a novel version of the Green function comparison theorem for data matrices with dependent entries. En route to proving edge universality, we establish that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Marcenko–Pastur law uniformly up to the edges of the spectrum with an error of order $(N\eta)^{-1}$ where $\eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. Combining these results with existing techniques we also show bulk universality of covariance matrices. All our results hold for both real and complex valued entries.

Functional CLT of eigenvectors for large sample covariance matrices

In order to investigate property of the eigenvector matrix of sample covariance matrix $$\mathbf {S}_n$$ , in this paper, we establish the central limit theorem of linear spectral statistics associated with a new form of empirical spectral distribution $$H^{\mathbf {S}_n}$$ , based on eigenvectors and eigenvalues of sample covariance matrix $$\mathbf {S}_n$$ . Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of $$H^{\mathbf {S}_n}$$ , indexed by a set of functions with continuous third order derivatives over an interval including the support of Marcenko–Pastur law. This result provides further evidences to support the conjecture that the eigenmatrix of sample covariance matrix is asymptotically Haar distributed.

Eigenvalue Estimation of Hyperspectral Wishart Covariance Matrices From Limited Number of Samples

Estimation of covariance matrices is a fundamental step in hyperspectral remote sensing where most detection algorithms make use of the covariance matrix in whitening procedures. We present a simple method to estimate all p eigenvalues of a Wishart-distributed sampled covariance matrix (with which an improved covariance can be constructed) when the number of samples (n) is small, n/p >; 1 and less than a few tens. Our method is based on the Marcenko-Pastur (M-P) law, theory of eigenvalue bounds, and energy conservation. We compute an apparent multiplicity for each sampled eigenvalue and then shift the sampled eigenvalues according the maximum likelihood location (M-P mode). We impose energy conservation in two distinct regions; small eigenvalues and large eigenvalues, where the transition between the two regions is found by solving successive first-order regression equation for the sampled data. The method also improves the condition number of the data (small eigenvalues are shifted upward in values), hence, it is also “regularization,” where the regularization is a multiplicative vector regularization as opposed to the traditional additive scalar regularization where all eigenvalues are shifted upward by the same value.

On Capacity Scaling of Multi-Antenna Multi-Hop Networks: The Significance of the Relaying Strategy in the “Long Network Limit”

The sum-capacity <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$C$</tex></formula> of a static uplink channel with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex> </formula> single-antenna sources and an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> -antenna destination is known to scale linearly in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> , if the random channel matrix fulfills the conditions for the Marcenko-Pastur law: if each source transmits at power <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$P/n$</tex></formula> , there exists a positive <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$c_{0}$</tex></formula> , such that <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\lim_{n \ura{} \infty}C/n=c_{0}$</tex></formula> almost surely. This paper addresses the question to which extent this result carries over to multi-hop networks. Specifically, an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$L+1$</tex></formula> -hop network with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex> </formula> non-cooperative source antennas, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> fully cooperative destination antennas, and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L$</tex></formula> relay stages of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n_{\cal R}$</tex></formula> (cooperative or non-cooperative) relay antennas each is considered. Four relaying strategies are assessed based on the interrelationship between two sequences. For each considered strategy <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm XF}$</tex></formula> , there exists a sequence <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(c_{L}^{\rm XF})_{L=0}^{\infty}$</tex> </formula> , such that <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$c_{L}^{\rm XF}=\lim_{n \ura{} \infty}R^{\rm XF}_{L}/n$</tex></formula> almost surely, where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$R^{\rm XF}_{L}$</tex></formula> denotes the supremum of the set of sum-rates that are achievable by the strategy over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L\!+\!1$</tex> </formula> hops. This sequence depends on the sequence <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(P_{L})_{L=0}^{\infty}$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$P_{L}$</tex></formula> corresponds to the power of the source stage and each of the relay stages in an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L\!+\!1$</tex></formula> -hop network. Results are summarized as follows: <orderedlist continuation="restarts" numeration="bullet" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <listitem><para>Decode &amp; forward (DF): For <formula formulatype="inline"><tex Notation="TeX">$n_{\cal R}=n$</tex></formula>, <formula formulatype="inline"><tex Notation="TeX">$c^{\rm DF}_{L}$</tex></formula> is constant with respect to <formula formulatype="inline"> <tex Notation="TeX">$L$</tex></formula>, if also <formula formulatype="inline"><tex Notation="TeX">$P_{L}$</tex></formula> is constant with respect to <formula formulatype="inline"> <tex Notation="TeX">$L$</tex></formula>.</para></listitem></orderedlist>

Eigenvalues of a statistical mechanics matrix

We consider the problem of finding the spectrum of an n × n matrix which arises in the study of a certain model of long-range interactions in a one-dimensional statistical mechanics system. Our analysis exhibits a curious resemblance of the suitably normalized distribution of eigenvalues to the Marcenko–Pastur law in the limit n → ∞.

Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States

We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in (C^d)^{otimes k}, where k and p/d^k are fixed while d grows. When k=1, the Marcenko-Pastur law determines (up to small corrections) not only the largest eigenvalue ((1+sqrt{p/d^k})^2) but the smallest eigenvalue (min(0,1-sqrt{p/d^k})^2) and the spectral density in between. We use the method of moments to show that for k>1 the largest eigenvalue is still approximately (1+sqrt{p/d^k})^2 and the spectral density approaches that of the Marcenko-Pastur law, generalizing the random matrix theory result to the random tensor case. Our bound on the largest eigenvalue has implications both for sampling from a particular heavy-tailed distribution and for a recently proposed quantum data-hiding and correlation-locking scheme due to Leung and Winter. Since the matrices we consider have neither independent entries nor unitary invariance, we need to develop new techniques for their analysis. The main contribution of this paper is to give three different methods for analyzing mixtures of random product states: a diagrammatic approach based on Gaussian integrals, a combinatorial method that looks at the cycle decompositions of permutations and a recursive method that uses a variant of the Schwinger-Dyson equations.

Network and eigenvalue analysis of financial transaction networks

We study a dataset containing all financial transactions between the accounts of practically all major financial players within Austria over one year. We empirically analyze transaction networks of money (in and out) flows and report the characteristic network parameters. We observe a significant dependence of network topology on the time scales of observation, and remarkably low correlation between node degrees and transaction volume. We further use transaction timeseries of the financial agents to compute covariance matrices and their eigenvalue spectra. Eigenvectors corresponding to eigenvalues deviating from the Marcenko-Pastur law are analyzed in detail. The potential for practical use as an automated detection mechanism for abnormal behavior of financial players is discussed. The opinion expressed in this paper is that of the authors and does not necessarily reflect the opinion of the OeNB or the ESCB.

Tyler's M-Estimator, Random Matrix Theory, and Generalized Elliptical Distributions with Applications to Finance

In recent publications standard methods of random matrix theory have been applied to principal components analysis of high-dimensional financial data. We discuss the fundamental results and potential shortcomings of random matrix theory in the light of the stylized facts of empirical finance. It is shown that the Marcenko-Pastur law generally fails when analyzing the empirical distribution function of the eigenvalues given by the sample covariance matrix of generalized elliptically distributed data. As an alternative we derive a random matrix referred to as the 'spectral estimator' which is distribution-free within the class of generalized elliptical distributions. We show that the spectral estimator corresponds to Tyler's M-estimator. Substituting the sample covariance matrix by the spectral estimator resolves the problems which are due to the stylized facts and the Marcenko-Pastur law remains valid. This holds even if the data are not generalized elliptically distributed but mutually independent.

Cavity approach to the spectral density of sparse symmetric random matrices

The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally treelike, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, showing excellent agreement.

The Polynomial Method for Random Matrices

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marcenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.