Results In Random Matrix Theory Research Articles

ON BLOCKWISE AND REFERENCE PANEL-BASED ESTIMATORS FOR GENETIC DATA PREDICTION IN HIGH DIMENSIONS.

Genetic prediction holds immense promise for translating genetic discoveries into medical advances. As the high-dimensional covariance matrix (or the linkage disequilibrium (LD) pattern) of genetic variants often presents a block-diagonal structure, numerous methods account for the dependence among variants in predetermined local LD blocks. Moreover, due to privacy considerations and data protection concerns, genetic variant dependence in each LD block is typically estimated from external reference panels rather than the original training data set. This paper presents a unified analysis of blockwise and reference panel-based estimators in a high-dimensional prediction framework without sparsity restrictions. We find that, surprisingly, even when the covariance matrix has a block-diagonal structure with well-defined boundaries, blockwise estimation methods adjusting for local dependence can be substantially less accurate than methods controlling for the whole covariance matrix. Further, estimation methods built on the original training data set and external reference panels are likely to have varying performance in high dimensions, which may reflect the cost of having only access to summary level data from the training data set. This analysis is based on novel results in random matrix theory for block-diagonal covariance matrix. We numerically evaluate our results using extensive simulations and real data analysis in the UK Biobank.

Sparse Random Hamiltonians Are Quantumly Easy

A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. For this task, there is a well-studied quantum algorithm that performs quantum phase estimation on an initial trial state that has a non-negligible overlap with a low-energy state. However, it is notoriously hard to give theoretical guarantees that such a trial state can be prepared efficiently. Moreover, the heuristic proposals that are currently available, such as with adiabatic state preparation, appear insufficient in practical cases. This paper shows that, for most random sparse Hamiltonians, the maximally mixed state is a sufficiently good trial state, and phase estimation efficiently prepares states with energy arbitrarily close to the ground energy. Furthermore, any low-energy state must have non-negligible quantum circuit complexity, suggesting that low-energy states are classically nontrivial and phase estimation is the optimal method for preparing such states (up to polynomial factors). These statements hold for two models of random Hamiltonians: (i) a sum of random signed Pauli strings and (ii) a random signed d-sparse Hamiltonian. The main technical argument is based on some new results in nonasymptotic random matrix theory. In particular, a refined concentration bound for the spectral density is required to obtain complexity guarantees for these random Hamiltonians. Published by the American Physical Society 2024

Ensemble averaging in JT gravity from entanglement in Matrix Quantum Mechanics

We consider the generalization of a matrix integral with arbitrary spectral curve ρ0(E) to a 0+1D theory of matrix quantum mechanics (MQM). Using recent techniques for 1D quantum systems at large-N, we formulate a hydrodynamical effective theory for the eigenvalues. The result is a simple 2D free boson BCFT on a curved background, describing the quantum fluctuations of the eigenvalues around ρ0(E), which is now the large-N limit of the quantum expectation value of the eigenvalue density operator hat{rho}(E) .The average over the ensemble of random matrices becomes a quantum expectation value. Equal-time density correlations reproduce the results (including non-perturbative corrections) of random matrix theory. This suggests an interpretation of JT gravity as dual to a one-time-point reduction of MQM.As an application, we compute the Rényi entropy associated to a bipartition of the eigenvalues. We match a previous result by Hartnoll and Mazenc for the c = 1 matrix model dual to two-dimensional string theory and extend it to arbitrary ρ0(E). The hydrodynamical theory provides a clear picture of the emergence of spacetime in two dimensional string theory. The entropy is naturally finite and displays a large amount of short range entanglement, proportional to the microcanonical entropy. We also compute the reduced density matrix for a subset of n < N eigenvalues.

Research on computer network security protection technology based on a superiority algorithm

Abstract The rapid development of computer network technology has brought great convenience to all walks of life and made the transmission of various information faster. A secure and reliable computer network environment has a critical impact on the functioning of the computer network. In this paper, we derive and analyze the detection probability, false alarm probability and judgment threshold of the new algorithm based on the results of random matrix theory based on the feature-fitting superiority algorithm, which can exhibit better detection performance. In the framework of matrix theory, the superiority algorithm reduces the problem of network security to some extent and improves the performance relative to other fitted superiority detection algorithms. For common threats to network security, give the corresponding ideas to deal with network security issues from a technical perspective. The detection rate and false alarm rate are calculated by testing the total number, the number of intrusion bars, the number of detected intrusion bars, and the number of false alarm bars detected by the system, respectively. The test results show that the average detection rate is still greater than 94%, the average false alarm rate has improved, and the average false alarm rate is controlled at about 3.1%, which can fully meet the system requirements. Give reference to further enhance the reliability and stability of network security systems.

Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio

In this paper, new results in random matrix theory are derived which allow us to construct a shrinkage estimator of the global minimum variance (GMV) portfolio when the shrinkage target is a random object. More specifically, the shrinkage target is determined as the holding portfolio estimated from previous data. The theoretical findings are applied to develop theory for dynamic estimation of the GMV portfolio, where the new estimator of its weights is shrunk to the holding portfolio at each time of reconstruction. Both cases with and without overlapping samples are considered in the paper. The non-overlapping samples corresponds to the case when different data of the asset returns are used to construct the traditional estimator of the GMV portfolio weights and to determine the target portfolio, while the overlapping case allows intersections between the samples. The theoretical results are derived under weak assumptions imposed on the data-generating process. No specific distribution is assumed for the asset returns except from the assumption of finite $4+\varepsilon$, $\varepsilon>0$, moments. Also, the population covariance matrix with unbounded spectrum can be considered. The performance of new trading strategies is investigated via an extensive simulation. Finally, the theoretical findings are implemented in an empirical illustration based on the returns on stocks included in the S\&P 500 index.

Eigenvalue spectra and stability of directed complex networks.

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction network between components has on the eigenvalue spectrum. We build on previous results, which usually only take into account the mean degree of the network, by allowing for nontrivial network degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix of a general weighted and directed network. Using these results, which are valid for any large well-connected complex network, we then derive compact formulas for the corrections (due to nonzero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semicircle law, the Girko circle law, and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat analytical expression for the eigenvalue density of a directed Barabási-Albert network. We are thus able to deduce that network heterogeneity is mostly a destabilizing influence in complex dynamical systems.

Onset of universality in the dynamical mixing of a pure state

We study the time dynamics of random density matrices generated by evolving the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians. We show that the spectral statistics of the resulting mixed state is well described by random matrix theory (RMT) and undergoes a crossover from the GOE to the Gaussian unitary ensemble (GUE) for short and large times respectively. Using a semi-analytical treatment relying on a power series of the density matrix as a function of time, we find that the crossover occurs in a characteristic time that scales as the inverse of the Hilbert space dimension. The RMT results are contrasted with a paradigmatic model of many-body localization in the chaotic regime, where the GUE statistics is reached at large times, while for short times the statistics strongly depends on the peculiarity of the considered subspace.

A Study of Intelligent Paper Grouping Model for Adult Higher Education Based on Random Matrix

This paper presents a comprehensive study and analysis of the intelligent grouping of papers in adult higher education using a random matrix approach. Using the results of random matrix theory on the eigenvalues of the sample covariance matrix, the energy of each subspace is estimated, and the estimated energy is then used to construct a subspace weighting matrix. The statistical properties of the sample covariance matrix eigenvectors are analyzed using the first-order perturbation approximation, and then, asymptotic results from random matrix theory on the projection of the sample covariance matrix signal subspace to the real signal parametrization are used to obtain the weighting matrix based on the random matrix eigenvectors. Dynamic adjustment according to the fitness of individuals in the population is performed to ensure population diversity, while the combination of the small habitat technique can avoid the algorithm from falling into early convergence. The algorithm introduces chaos theory to optimize the population initialization process and uses the dynamic traversal randomness of chaos to select individuals in the population so that the initial population is close to the desired target solution. The design of the fitness function in the genetic algorithm generally maps the objective function of the problem to the fitness function. A good fitness function can directly reflect the quality of the individuals in the group. Based on the in-depth study of the basic attributes of the test questions and the principles of test paper evaluation, the mathematical model and objective function of intelligent paper grouping are determined by the difficulty, knowledge points, and cognitive level of the test questions as the main constraints, and NCAGA is applied to the intelligent paper grouping method, which better completes the intelligent paper grouping session for the computer system architecture course. In the process of designing the intelligent grouping algorithm, for the situations of premature convergence and convergence to locally optimal solutions that easily occur in the traditional genetic algorithm, this paper adopts the approach of adaptive adjustment of crossover probability and variation probability to improve the algorithm and achieves satisfactory results. Based on extensive business research, this paper completes the requirement analysis of the online practice system based on the intelligent grouping of papers and presents the functional design and database design of the key functional modules in the system in detail. Finally, this paper conducts functional tests on the system and analyses the test results.

Development Model of College English Education Service Talents Based on Nonlinear Random Matrix

The development of artificial intelligence makes people’s life and work easier and more effective, and computer-based online exams and marking not only improve students’ learning efficiency but also reduce the pressure of teachers’ marking work. For objective questions, marking has gone from manual marking to cursor reader marking to computerized character matching, and the correct rate of marking has soared to 100%; for subjective questions, foreign systems such as PEG and E-rater have been used, and domestic systems such as those using English large corpus similarity matching and those based on natural language understanding using intelligent algorithms have been used for marking. Most of these systems are based on some shallow linguistic features such as rules and LSAs for marking, and there is no deep perception of English language sense. Although the current intelligent marking systems have made a lot of achievements, they do not fundamentally solve the problem of the rationality of intelligent marking of subjective questions. In this article, we propose a regularized discriminant analysis algorithm with good estimation of the mean, and a dimensionality reduction algorithm for high-dimensional missing data by using the relevant research results of random matrix theory to address the problems of traditional machine learning methods in high-dimensional data analysis. Although the linear discriminant analysis algorithm performs well in solving many practical problems, it works poorly in dealing with high-dimensional data. The specific analysis is as follows: in terms of age characteristics, the mobile population under the age of 35 has a significant preference for urban consumer comfort, and it increases with age, peaking at the stage of 30–35 years old and then decreasing rapidly. For this reason, a regularized discriminant analysis algorithm based on random matrix theory is proposed. First, a good estimate of the high-dimensional covariance matrix is made by the nonlinear shrinkage method or the eigenvalue interception method, respectively; then, the estimated high-dimensional covariance matrix is used to calculate the discriminant function values and perform the classification. The classification experiments conducted on simulated and real datasets show that the proposed algorithm is not only more widely applicable but also has a high correct classification rate.

Research on the Nonlinear Random Matrix Model for Urban Information Model and Modern Urban Governance

In this paper, a nonlinear stochastic matrix approach is used to conduct an in-depth study and analysis of modern urban governance, and an urban information model is designed for practical application based on it. Random matrix theory can calculate and give the average result of all interactions contained in a complex system. By comparing the different characteristics of statistical properties of real systems and random matrices, the special nonrandom properties of real systems are deduced. The dissipative structure theory of open complex systems, the evolution of network entropy describing the robustness of complex networks, can explain the empirical results of random matrix theory from the point of view of system evolution; through the open development of fair data and prevention tools to achieve algorithm correction, promote the design optimization of intelligent models, intelligent platforms, and intelligent programs; and commit to the benign development of technology. At the institutional level, it is necessary to keep the smart city construction synchronized with the rule of law construction, with a two-pronged approach of policy incentives and legal constraints, in addition to the overall promotion of government transformation and pluralistic governance patterns. It exceeds the carrying capacity and adaptability of social systems and institutions. In fact, the productive forces with science and technology as the core are the most active forces in social change and development. The urban information model is a technical and conceptual change triggered by the wave of information in the field of urban planning and construction and urban management, which takes the 3D model as a carrier to realize digital information application and unification in the whole life cycle of territorial spatial planning, urban design, and engineering and construction projects through the simulation of spatial planning and urban design spatial elements as well as the entry, management, and application of engineering and construction project information. Digital technology is applied in the whole life cycle of land spatial planning, urban design, and engineering projects to achieve fine governance and efficient management of urban planning.

Reliability Analysis of Sports Training Evaluation Index Based on Random Matrix

This paper presents an in-depth study and analysis of the reliability of indicators for the evaluation of sports training using the algorithm of random matrices. Random matrix theory is used to obtain the degree of randomness of the data and the overall correlation characteristics of the data through statistical analysis of the energy spectrum and eigenstates of complex systems. Using the results of random matrix theory on the eigenvalues of the sample covariance matrix, the energy of each subspace is estimated, and the estimated energy is then used to construct the subspace weighting matrix. The problem of reliable finite-time stability of the model under several types of uncertain time-lagged bounded transfer probability Markov jump parameter compatibility indicators is also explored. The four basic theoretical issues, namely, value orientation, construction basis, construction principles, and conceptual model of the sports training work evaluation index system, are studied. This paper takes the construction of a sports training evaluation index system as the research object and uses the literature method, expert interview method, questionnaire survey method, data analysis method, Delphi method, and hierarchical analysis method as the main research methods to construct a set of sports training evaluation index system as the research task, to enrich the idea of sports training evaluation work, and to provide a reference for the future government to be able to develop a sports training evaluation index system that is in line with the development of the times. The research method was used to collect the opinions of experts. The subjective opinions given by experts were collected using the assignment method, and the weight coefficients of each index were calculated using the AHP hierarchical analysis method. The result is that the index of basic conditions (0.3591) has the largest weight in the whole system. After that, organization and management (0.1911), work effectiveness (0.1376), staffing (0.103), activity development (0.0828), financial and fund management (0.0562), publicity and reporting (0.0382), and material provision (0.032) are in order. The evaluation results of the index system are consistent with the current actual situation of specialization teaching, indicating that the index system for evaluating the effectiveness of high school sports training has good validity.

A Model for Teaching English Translation in Universities Based on the Random Matrix Optimization Bell Translation Model

This paper adopts the random matrix optimization fusion of Bell’s translation model to conduct in-depth research and analysis on the teaching model of English translation in colleges and universities. Bell’s translation model theory explicates the mental model of the translation process to a certain extent based on psycholinguistic research results. Using the results of random matrix theory on the eigenvalues of the sample covariance matrix, the energy of each subspace is estimated, and the estimated energy is then used to construct the subspace-weighting matrix. The statistical properties of the sample covariance matrix eigenvectors were analyzed using the first-order perturbation approximation, and then asymptotic results from random matrix theory on the projection of the sample covariance matrix signal subspaces to the real signal parametrization were used to obtain the weighting matrix based on the random matrix eigenvectors. The analysis shows that the teacher can help students understand the teaching content and master the knowledge points by using guided meta-discoursal to indicate the logical relationships inherent in linguistic expressions and make the discourse transitions natural, coherent, and organized. The use of interactive meta-discoursal attracts students’ attention and treats them as participants in the discourse, making them feel a sense of belonging and satisfying their need for psychological belonging, which is beneficial to the students’ main position in the classroom. In teaching methods, there is a principle of comprehensible input, which is to provide students with a lot of interesting and understandable listening and reading input. In the translation classroom, the translation teachers code-switched infrequently and mainly used three types of code-switching: inter-sentence code-switching, intra-sentence code-switching, and additional code-switching to carry out translation teaching. Among them, intra-sentence code-switching accounts for 57.46%, inter-sentence code-switching accounts for 41.53%, and the least frequent is additional code-switching, accounting for 1.01%.

Random Matrix Theory Predictions of Dominant Mode Rejection Beamformer Performance

Adaptive beamformers use a sensor covariance matrix estimated from data snapshots to mitigate directional interference and attenuate uncorrelated noise. Dominant mode rejection (DMR) is a variant of the classic minimum variance distortionless response (MVDR) algorithm that replaces the smallest eigenvalues in the covariance estimate by their average while leaving the largest ones unmodified. Since DMR does not invert the small eigenvalues of the sample covariance, it achieves a higher white noise gain than MVDR while still suppressing loud interferers, thereby yielding a higher signal-to-interference-plus-noise ratio (SINR). MVDR achieves an average output SINR within 3&#x00A0;dB of the SINR achievable with the true covariance with approximately twice as many snapshots as sensors. Prior empirical analyses showed that achieving the same SINR with DMR only requires approximately twice as many snapshots as interferers. This paper derives analytical models of white noise gain and interference leakage for DMR using random matrix theory (RMT), specifically the eigenvalue and eigenvector limiting results of the spiked covariance model. Applying the new analytical models confirms the empirical DMR snapshot requirement. To handle finite cases with a large number of interferers relative to the array size, this paper derives a modified spiked covariance model that improves the accuracy of RMT results, and hence DMR performance predictions.

Breakdown of random matrix universality in Markov models.

Biological systems need to react to stimuli over a broad spectrum of timescales. If and how this ability can emerge without external fine-tuning is a puzzle. We consider here this problem in discrete Markovian systems, where we can leverage results from random matrix theory. Indeed, generic large transition matrices are governed by universal results, which predict the absence of long timescales unless fine-tuned. We consider an ensemble of transition matrices and motivate a temperature-like variable that controls the dynamic range of matrix elements, which we show plays a crucial role in the applicability of the large matrix limit: as the dynamic range increases, a phase transition occurs whereby the random matrix theory result is avoided, and long relaxation times ensue, in the entire "ordered" phase. We furthermore show that this phase transition is accompanied by a drop in the entropy rate and a peak in complexity, as measured by predictive information [Bialek, Nemenman, and Tishby Neural Comput. 13, 2409 (2001)NEUCEB0899-766710.1162/089976601753195969]. Extending the Markov model to a Hidden Markov model (HMM), we show that observable sequences inherit properties of the hidden sequences, allowing HMMs to be understood in terms of more accessible Markov models. We then apply our findings to fMRI data from 820 human subjects scanned at wakeful rest. We show that the data can be quantitatively understood in terms of the random model, and that brain activity lies close to the phase transition when engaged in unconstrained, task-free cognition-supporting the brain criticality hypothesis in this context.

Random Matrix Theory and Its Applications

This article reviews the important ideas behind random matrix theory (RMT), which has become a major tool in a variety of disciplines, including mathematical physics, number theory, combinatorics and multivariate statistical analysis. Much of the theory involves ensembles of random matrices that are governed by some probability distribution. Examples include Gaussian ensembles and Wishart–Laguerre ensembles. Interest has centered on studying the spectrum of random matrices, especially the extreme eigenvalues, suitably normalized, for a single Wishart matrix and for two Wishart matrices, for finite and infinite sample sizes in the real and complex cases. The Tracy–Widom Laws for the probability distribution of a normalized largest eigenvalue of a random matrix have become very prominent in RMT. Limiting probability distributions of eigenvalues of a certain random matrix lead to Wigner’s Semicircle Law and Marc˘enko–Pastur’s Quarter-Circle Law. Several applications of these results in RMT are described in this article.

Generation of Bures-Hall mixed states using coupled kicked tops

We simulate a system of coupled kicked tops to generate random density matrices distributed according to the Bures-Hall measure, which has an important role in quantum information theory. We study the effects of stochasticity and coupling parameters of the coupled-kicked-top system on the behavior of the associated bipartite pure state, eigenvalues and eigenvectors of the reduced state, and average entropies. For strongly chaotic phase space and adequate coupling between the constituting tops (subsystems), we find that the results of simulation agree with analytical results of random matrix theory. We also examine local fluctuation properties of the Bures-Hall eigenvalues using the distribution of nearest-neighbor spacing ratios. In the limit of high-dimensional density matrices, the empirical ratio distribution is found to approach the Wigner-surmise-like result for the Gaussian unitary ensemble. Additionally, we present some closed-form results for the Bures-Hall spectral density and corresponding moments using Meijer $G$ functions.

Secrecy rate analysis for reconfigurable intelligent surface-assisted mimo communications with statistical CSI

In this paper, a reconfigurable intelligent surface (RIS)-assisted MIMO wireless secure communication system is considered, in which a base station (BS) equipped with multiple antennas exploits statistical channel state information to communicate with a legitimate multi-antenna user, in the presence of an eavesdropper, also equipped with multiple antennas. We firstly obtain an analytical expression of the ergodic secrecy rate based on the results of large-dimensional random matrix theory. Then, a jointly alternating optimization algorithm with the method of Taylor series expansion and the projected gradient ascent method is proposed to design the transmit covariance matrix at the BS, as well as the diagonal phase- shifting matrix to maximize the ergodic secrecy rate. Simulations are conducted to demonstrate the accuracy of the derived analytical expressions, as well as the superior performance of our proposed algorithm.

Using singular value distribution of backscattered ultrasound waves for tracking the pulmonary edema caused by COVID-19 in lung: A phantom-based study

A condition caused by COVID-19, pulmonary edema is an abnormal buildup of fluid in lungs. B-lines tracking of ultrasound images is a common way to track lung abnormality, however due to multiple scattering this method tracks the air volume change only on the lung surface. The aim of this phantom-based study is to assess the pulmonary edema in lungs using multiple scattering of ultrasonic waves in MHz range. A 128-element array transducer is used to form an Inter-elements Response Matrix (IRM) by firing a 5.2 MHz broadband pulse, element by element and receiving the backscattered signal by the whole array. The distribution of singular values of IRM in frequency domain is then analyzed for polymer sponges (n = 4) with different Water Volume Content (WVC) ranging from 5 to 25 ml mimicking pulmonary edema. The results of Random Matrix Theory are used to show that the mode of the Probability Density Function of singular values increases as the WVC increases in sponge indicating the change in scattering regime. The B-mode image of sponges with different WVC did not represent neither A nor B lines, however our results shows that there is a correlation between the WVC in sponge and the strongest singular value of IRM (R = 0.89) indicating the potential of using quantitative ultrasound in lung for early diagnosis of COVID-19.

Spectral fluctuations in the Sachdev-Ye-Kitaev model

We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings for the number variance or exponentially short times in the case of the spectral form factor.

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

We consider a family of tronquée solutions of the Painlevé II equationq″(s)=2q(s)3+sq(s)−(2α+12),α>−12, which is characterized by the Stokes multiplierss1=−e−2απi,s2=ω,s3=−e2απi with ω being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if ω=0. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for α>−1/2 and ω≥0, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters α and ω are chosen to be special values. Some applications of our results in random matrix theory are also discussed.