Application Of Random Matrix Theory Research Articles

Fading ergodicity

Eigenstate thermalization hypothesis (ETH) represents a breakthrough in many-body physics since it allows us to link thermalization of physical observables with the applicability of random matrix theory (RMT). Recent years were also extremely fruitful in exploring possible counterexamples to thermalization, ranging, among others, from integrability, single-particle chaos, many-body localization, many-body scars, to Hilbert-space fragmentation. In all these cases the conventional ETH is violated. However, it remains elusive how the conventional ETH breaks down when one approaches the boundaries of ergodicity, and whether the range of validity of the conventional ETH coincides with the validity of RMT-like spectral statistics. Here we bridge this gap and we introduce a scenario of the ETH breakdown in many-body quantum systems, dubbed fading ergodicity regime, which establishes a link between the conventional ETH and nonergodic behavior. We conjecture this scenario to be relevant for the description of finite many-body systems at the boundaries of ergodicity, and we provide numerical and analytical arguments for its validity in the quantum sun model of ergodicity-breaking phase transition. For the latter, we provide evidence that the breakdown of the conventional ETH is not associated with the breakdown of the RMT-like spectral statistics. Published by the American Physical Society 2024

Optimal covariance cleaning for heavy-tailed distributions: Insights from information theory.

In optimal covariance cleaning theory, minimizing the Frobenius norm between the true population covariance matrix and a rotational invariant estimator is a key step. This estimator can be obtained asymptotically for large covariance matrices, without knowledge of the true covariance matrix. In this study, we demonstrate that this minimization problem is equivalent to minimizing the loss of information between the true population covariance and the rotational invariant estimator for normal multivariate variables. However, for Student's t distributions, the minimal Frobenius norm does not necessarily minimize the information loss in finite-sized matrices. Nevertheless, such deviations vanish in the asymptotic regime of large matrices, which might extend the applicability of random matrix theory results to Student's t distributions. These distributions are characterized by heavy tails and are frequently encountered in real-world applications such as finance, turbulence, or nuclear physics. Therefore, our work establishes a connection between statistical random matrix theory and estimation theory in physics, which is predominantly based on information theory.

Application of random matrix theory combined with the singular value decomposition to journal bearings uncertainty analysis

Application of random matrix theory combined with the singular value decomposition to journal bearings uncertainty analysis

High Dimensional Normality of Noisy Eigenvectors

We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing the stochastic eigenstate equation (SEE) (Bourgade and Yau in Comm Math Phys, 2013) which describes the Lie group valued flow of eigenvectors induced by matrix valued Brownian motion. We consider the associated colored eigenvector moment flow defining an SDE on a particle configuration space. This flow extends the eigenvector moment flow first introduced in Bourgade and Yau (Comm Math Phys, 2013) to the multicolor setting. However, it is no longer driven by an underlying Markov process on configuration space due to the lack of positivity in the semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged decay and contractive properties. This allows us to establish optimal time of relaxation to equilibrium for the colored eigenvector moment flow and prove joint asymptotic normality for eigenvectors. Applications in random matrix theory include the explicit computations of joint eigenvector distributions for general Wigner type matrices and sparse graph models when corresponding eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector distributions for Lévy matrices when the eigenvectors correspond to small energy levels.

A Study on the Application of Random Matrix Theory in the Construction of the Evaluation System of Public English Flipped Classroom Teaching in Higher Education Institutions

With the development of science and technology, the advent of the knowledge era, the Internet era, the WeChat era, and the mobile era as well as the development of multimedia have brought about great changes in the dissemination, communication, construction, and sharing of knowledge. This series of changes has led to unprecedented changes in teaching philosophy, teaching mode, teaching objectives, teaching design, teaching methods, and teaching approaches in higher vocational education. With the deepening of the new round of curriculum reform, flipped classroom teaching has been generally accepted and widely used. Therefore, it is essential to evaluate flipped classroom teaching. Based on the introduction of the random matrix theory, this paper attempts to establish a complete and scientific evaluation index system in conjunction with the flipped public English classroom in higher education institutions, so as to provide students and teachers with a reliable means of evaluation and thus promote the practical application of English teaching in higher education.

The geometry of evolved community matrix spectra

Random matrix theory has been applied to food web stability for decades, implying elliptical eigenvalue spectra and that large food webs should be unstable. Here we allow feasible food webs to self-assemble within an evolutionary process, using simple Lotka–Volterra equations and several elementary interaction types. We show that, as complex food webs evolve under {10^5} invasion attempts, the community matrix spectra become bi-modal, rather than falling onto elliptical geometries. Our results raise questions as to the applicability of random matrix theory to the analysis of food web steady states.

Application of Random Matrix Theory With Maximum Local Overlapping Semicircles for Comorbidity Analysis

In December 2019, the COVID-19 pandemic began, which has claimed the lives of millions of people around the world. This article presents a regional analysis of COVID-19 in Mexico. Due to comorbidities in Mexican society, this new pandemic implies a higher risk for the population. The study period runs from 12 April to 5 October 2020 761,665. This article proposes a unique methodology of random matrix theory in the moments of a probability measure that appears as the limit of the empirical spectral distribution by Wigner's semicircle law. The graphical presentation of the results is done with Machine Learning methods in the SuperHeat maps. With this, it was possible to analyze the behavior of patients who tested positive for COVID-19 and their comorbidities, with the conclusion that the most sensitive comorbidities in hospitalized patients are the following three: COPD, Other Diseases, and Renal Diseases.

High-order correlations in species interactions lead to complex diversity-stability relationships for ecosystems.

How ecosystems maintain stability is an active area of research. Inspired by applications of random matrix theory in nuclear physics, May showed decades ago that in an ecosystem model with many randomly interacting species, increasing species diversity decreases the stability of the ecosystem. There have since been many additions to May's efforts, one being an improved understanding the effect of mutualistic, competitive, or predator-prey-like correlations between pairs of species. Here we extend a random matrix technique developed in the context of spin-glass theory to study the effect of high-order correlations among species interactions. The resulting analytically solvable models include next-to-nearest-neighbor correlations in the species interaction network, such as the enemy of my enemy is my friend, as well as higher-order correlations. We find qualitative differences from May and others' models, including nonmonotonic diversity-stability relationships. Furthermore, inclusion of particular next-to-nearest-neighbor correlations in predator-prey as opposed to mutualist-competitive networks causes the former to transition to being more stable at higher species diversity. We discuss potential applicability of our results to microbiota engineering and to the ecology of interpredator interactions, such as cub predation between lions and hyenas as well as companionship between humans and dogs.

Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

We prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When the underlying Lévy process is non-Gaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operator-valued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operator-valued ^*-probability spaces, where the limits of sample covariance matrices play the role of non-commutative Lévy processes whose increments are free with amalgamation.

Appearance of Random Matrix Theory in deep learning

We investigate the local spectral statistics of the loss surface Hessians of artificial neural networks, where we discover agreement with Gaussian Orthogonal Ensemble statistics across several network architectures and datasets. These results shed new light on the applicability of Random Matrix Theory to modelling neural networks and suggest a role for it in the study of loss surfaces in deep learning.

Applications in random matrix theory of a PIII′ τ-function sequence from Okamoto’s Hamiltonian formulation

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

Undesired Cross Terms [Lecture Notes

This lecture note briefly describes two examples in mathematics on how and why cross terms are disliked: orthogonal space-time block code (OSTBC) from orthogonal designs and free probability theory. The first example shows that the elimination of cross terms can significantly simplify the optimization that has many practical applications in, for instance, multiple antenna systems in wireless communications. The second example gives an intuitive and/or fundamental understanding of free probability theory, which has important applications in random matrix theory.

Nonlinear random matrix theory for deep learning**This article is an updated version of: Ha J-S, Park Y-J, Chae H-J, Park S-S and Choi H-L 2017 Nonlinear random matrix theory for deep learning Advances in Neural Information Processing Systems 30 ed I Guyon, U V Luxburg, S Bengio, H Wallach, R Fergus, S Vishwanathan and R Garnett (Red Hook, NY: Curran

Neural network configurations with random weights play an important role in the analysis of deep learning. They define the initial loss landscape and are closely related to kernel and random feature methods. Despite the fact that these networks are built out of random matrices, the vast and powerful machinery of random matrix theory has so far found limited success in studying them. A main obstacle in this direction is that neural networks are nonlinear, which prevents the straightforward utilization of many of the existing mathematical results. In this work, we open the door for direct applications of random matrix theory to deep learning by demonstrating that the pointwise nonlinearities typically applied in neural networks can be incorporated into a standard method of proof in random matrix theory known as the moments method. The test case for our study is the Gram matrix FFT, , where W is a random weight matrix, X is a random data matrix, and is a pointwise nonlinear activation function. We derive an explicit representation for the trace of the resolvent of this matrix, which defines its limiting spectral distribution. We apply these results to the computation of the asymptotic performance of single-layer random feature networks on a memorization task and to the analysis of the eigenvalues of the data covariance matrix as it propagates through a neural network. As a byproduct of our analysis, we identify an intriguing new class of activation functions with favorable properties.

A new stochastic computing paradigm for nonlinear Painlevé II systems in applications of random matrix theory

The aim of the present work is to investigate the stochastic numerical solutions of nonlinear Painleve II systems arising from studies of two-dimensional Yang-Mills theory, growth processes through fluctuation formulas in statistical physics, soft-edge random matrix distributions using the strength of bio-inspired heuristics through artificial neural networks (ANNs), genetic algorithm (GA)-based evolutionary computations and interior-point techniques (IPTs). A new mathematical modelling of the system is formulated through ANNs by defining an error function that exactly satisfies the initial conditions. The weights of ANN models optimized through a memetic computing approach that is based on a global search with GAs, and IPTs are used for an efficient local search. The designed scheme is substantiated through comparative analysis with a fully explicit Range-Kutta numerical procedure on nonlinear Painleve II systems by taking different magnitudes of forcing factors. The accuracy and convergence of the proposed scheme are validated through statistics performed on large numbers of simulations.

Applications of random-matrix theory and nonparametric change-point analysis to three notable systemic crises

This paper studies association between changes in absorption ratio and aggregate market returns in three systemic crises across a broad class of assets. Time series of normalized eigenvalue estimates reveal that crises are characterized by a general breakdown of correlation structure. The structure of return correlations is nonlinear and nonstationary across dierent asset groups. So we introduce a nonparametric technique to monitor divergence in distributions underlying successive observations of normalized dominant eigenvalue of the returns. Periods of high divergence imply a change in the correlation structure of asset returns. They are found to either precede or coincide with systemic shocks. An additional parametric analysis is provided as an informal check on the results obtained in the paper.

Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory

In this paper, we consider the asymptotic behavior of Xfn(n)≔∑i=1nfn(xi), where xi,i=1,…,n form orthogonal polynomial ensembles and fn is a real-valued, bounded measurable function. Under the condition that VarXfn(n)→∞, the Berry-Esseen (BE) bound and Cramer type moderate deviation principle (MDP) for Xfn(n) are obtained by using the method of cumulants. As two applications, we establish the BE bound and Cramer type MDP for linear spectrum statistics of Wigner matrix and sample covariance matrix in the complex cases. These results show that in the edge case (which means fn has a particular form f(x)I(x≥θn) where θn is close to the right edge of equilibrium measure and f is a smooth function), Xfn(n) behaves like the eigenvalues counting function of the corresponding Wigner matrix and sample covariance matrix, respectively.

LLN for Quadratic Forms of Long Memory Time Series and Its Applications in Random Matrix Theory

We show that a weak concentration property for quadratic forms of isotropic random vectors ${\bf x}$ is necessary and sufficient for the validity of the Marchenko-Pastur theorem for sample covariance matrices of random vectors having the form $C{\bf x}$, where $C$ is any rectangular matrix with orthonormal rows. We also obtain some general conditions guaranteeing the weak concentration property.

Modeling of Integration Processes of Ukraine to EU Using Random Matrix Theory

The application of random matrix theory made it possible to determine the place of Ukraine in separate clusters for certain groups of goods, as well as to find out the fact that Ukraine either operates within the general dynamics of the relevant market, or affects individual markets in some way in these groups of goods. It shows the close connection of our country with other subjects of the European region. According to the study results of ITC database, we can make a conclusion that the main share in Ukraine's foreign trade activities with the countries of Europe is taken by such groups of products as mill products, animal and vegetable fats, footwear, glass and its articles. At the same time, Ukraine is an important subject of the European market for inorganic chemistry products, wood and its products, paper and its derivatives, iron, steel, copper, aluminum, etc. We consider that in modern conditions, it is expedient for Ukraine to transfer from the export of raw materials and products of primary processing to the production and supply of the results of manufacturing industry, i.e. finished goods, to the European market. Finished goods, in their turn, contain a larger share of added value, and, therefore, can provide an increase in revenues to the state budget at the expense of exports, growth of volumes and value of GDP. At the same time, it will help to create additional job positions in the domestic labour market and increase the incomes of the population.

Distributed sequence memory of multidimensional inputs in recurrent networks

Recurrent neural networks (RNNs) have drawn interest from machine learning researchers because of their effectiveness at preserving past inputs for time-varying data processing tasks. To understand the success and limitations of RNNs, it is critical that we advance our analysis of their fundamental memory properties. We focus on echo state networks (ESNs), which are RNNs with simple memoryless nodes and random connectivity. In most existing analyses, the short-term memory (STM) capacity results conclude that the ESN network size must scale linearly with the input size for unstructured inputs. The main contribution of this paper is to provide general results characterizing the STM capacity for linear ESNs with multidimensional input streams when the inputs have common low-dimensional structure: sparsity in a basis or significant statistical dependence between inputs. In both cases, we show that the number of nodes in the network must scale linearly with the information rate and poly-logarithmically with the input dimension. The analysis relies on advanced applications of random matrix theory and results in explicit non-asymptotic bounds on the recovery error. Taken together, this analysis provides a significant step forward in our understanding of the STM properties in RNNs.

Normal Vector of a Random Hyperplane

Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some randomness assumption on the v_i. Our result has applications in random matrix theory. Consider an n by n random matrix with iid entries. We first prove an exponential bound on the upper tail for the least singular value, improving the earlier linear bound by Rudelson and Vershynin. Next, we derive optimal delocalization for the eigenvectors corresponding to eigenvalues of small modulus.