Symmetric Random Matrix Research Articles

The singularity probability of a random symmetric matrix is exponentially small

Let A A be drawn uniformly at random from the set of all n × n n\times n symmetric matrices with entries in { − 1 , 1 } \{-1,1\} . We show that \[ P ( det ( A ) = 0 ) ⩽ e − c n , \mathbb {P}( \det (A) = 0 ) \leqslant e^{-cn}, \] where c > 0 c>0 is an absolute constant, thereby resolving a long-standing conjecture.

The least singular value of a random symmetric matrix

Abstract Let A be an $n \times n$ symmetric matrix with $(A_{i,j})_{i\leqslant j}$ independent and identically distributed according to a subgaussian distribution. We show that $$ \begin{align*}\mathbb{P}(\sigma_{\min}(A) \leqslant \varepsilon n^{-1/2} ) \leqslant C \varepsilon + e^{-cn},\end{align*} $$ where $\sigma _{\min }(A)$ denotes the least singular value of A and the constants $C,c>0 $ depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$ . Along the way, we prove that the probability that A has a repeated eigenvalue is $e^{-\Omega (n)}$ , thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777–816].

Approximate Message Passing for sparse matrices with application to the equilibria of large ecological Lotka–Volterra systems

This paper is divided into two parts. The first part is devoted to the study of a class of Approximate Message Passing (AMP) algorithms which are widely used in the fields of statistical physics, machine learning, or communication theory. The AMP algorithms studied in this part are those where the measurement matrix has independent elements, up to the symmetry constraint when this matrix is symmetric, with a variance profile that can be sparse. The AMP problem is solved by adapting the approach of Bayati, Lelarge, and Montanari (2015) to this matrix model.The Lotka–Volterra (LV) model is the standard model for studying the dynamical behavior of large dimensional ecological food chains. The second part of this paper is focused on the study of the statistical distribution of the globally stable equilibrium vector of a LV system in the situation where the random symmetric interaction matrix among the living species is sparse, and in the regime of large dimensions. This equilibrium vector is the solution of a Linear Complementarity Problem, which distribution is shown to be characterized through the AMP approach developed in the first part. In the large dimensional regime, this distribution is close to a mixture of a large number of truncated Gaussians.

Efficient Inverse Model for Cloud Computing and Information Based on the Symmetric Random Matrix

With the continuous development of cloud storage technology, cloud storage has become a new way of storage for both enterprises and individuals. The cloud environment can support users to share data, but it can lead to some malicious users accessing or modifying data by illegal means, so access control is an important way to protect user data. In this paper, we propose a traceable attribute encryption scheme based on signature authentication and based on the information efficient inverse model of an asymmetric random matrix, which can trace the user who leaks information. A user attribute revocation scheme based on improved attribute encryption is also proposed, which enables data owners to reduce computational overhead. In cloud storage access control, the workload of the data owner increases exponentially when facing a high number of user attribute revocation problems, which are jointly decided by the data owner and the authorization center. An error proportional allocation (EPA) calculation method is developed to achieve an optimal estimation of system parameters in the lattice cipher scheme; experimental results show that the scheme has the advantages of shorter parameters, efficient computation, and lower storage load, proving that the scheme is resistant to IND-SAS-CPA under the standard model-based assumption of determining LWE semantic security against IND-SAS-CPA attacks; and it is used in a cloud file sharing (CFS) service framework, which can make sensitive cloud data free from the risk of privacy leakage.

Multi-Ellipsoidal Extended Target Tracking With Variational Bayes Inference

In this work, we propose a novel extended target tracking algorithm, which is capable of representing a target or a group of targets with multiple ellipses. Each ellipse is modeled by an unknown symmetric positive-definite random matrix. The proposed model requires solving two challenging problems. First, the data association problem between the measurements and the sub-objects. Second, the inference problem that involves non-conjugate priors and likelihoods which needs to be solved within the recursive filtering framework. We utilize the variational Bayes inference method to solve the association problem and to approximate the intractable true posterior. The performance of the proposed solution is demonstrated in simulations and real-data experiments. The results show that our method outperforms the state-of-the-art methods in terms of accuracy with lower computational complexity.

Noise sensitivity for the top eigenvector of a sparse random matrix

We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let v be the top eigenvector of an N×N sparse random symmetric matrix with an average of d non-zero centered entries per row. We resample k randomly chosen entries of the matrix and obtain another realization of the random matrix with top eigenvector v[k]. Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove that, if d≥N2∕9, with high probability, when k≪N5∕3, the vectors v and v[k] are almost collinear and, on the contrary, when k≫N5∕3, the vectors v and v[k] are almost orthogonal. A similar result holds for the eigenvector associated to the second largest eigenvalue of the adjacency matrix of an Erdős-Rényi random graph with average degree d≥N2∕9.

On the permanent of a random symmetric matrix

Let \(M_{n}\) denote a random symmetric \(n\times n\) matrix, whose entries on and above the diagonal are i.i.d. Rademacher random variables (taking values \(\pm 1\) with probability 1/2 each). Resolving a conjecture of Vu, we prove that the permanent of \(M_{n}\) has magnitude \(n^{n/2+o(n)}\) with probability \(1-o(1)\). Our result can also be extended to more general models of random matrices. In our proof, we build on and extend some techniques introduced by Tao and Vu, studying the evolution of permanents of submatrices in a random matrix process.

Stability of large complex systems with heterogeneous relaxation dynamics

We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size n i (with respect to its equilibrium value): . The a i > 0’s are the intrinsic decay rates, J ij is a real symmetric (N × N) Gaussian random matrix and measures the strength of pairwise interaction between different species. Our goal is to study how inhomogeneities in the intrinsic damping rates a i affect the stability of this dynamical system. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = T c. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian motion (DBM), starting at the initial position a i and evolving in ‘time’ T. In the large N → ∞ limit, using this DBM picture, we are able to completely characterize T c for arbitrary density μ(a) of the a i ’s. For a specific flat configuration , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N, we also compute the large deviation properties of the probability of stability on the stable side T < T c using a Coulomb gas representation.

On the smallest singular value of symmetric random matrices

AbstractWe show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta &gt; 0$ ). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

Random Toeplitz matrices: The condition number under high stochastic dependence

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix [Formula: see text] of dimension [Formula: see text] under the existence of the moment generating function of the random entries is [Formula: see text] with probability [Formula: see text] for any [Formula: see text], [Formula: see text]. Moreover, if the random entries only have the second moment, the condition number satisfies [Formula: see text] with probability [Formula: see text]. Also, we analyze the condition number of a random symmetric circulant matrix [Formula: see text]. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix [Formula: see text] we establish [Formula: see text], where [Formula: see text] is the minimum singular value of the matrix [Formula: see text]. The matrix [Formula: see text] is a random circulant matrix and [Formula: see text], where [Formula: see text] are deterministic matrices, [Formula: see text] indicates the conjugate transpose of [Formula: see text] and [Formula: see text] are random diagonal matrices. From random experiments, we conjecture that [Formula: see text] is well-conditioned if the moment generating function of the random entries of [Formula: see text] exists.

Thermalization processes induced by quantum monitoring in multilevel systems.

We study the heat statistics of a multilevel N-dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number M of measurements (M→∞). In this context, the conditions allowing for an infinite-temperature thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the nonequilibrium evolution of the system and its initial state. Exceptions to ITT, which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasicommuting) with the Hamiltonian of the quantum system or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces (N→∞), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits M→∞ and N→∞ matters: When N is fixed and M diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A nontrivial result is obtained fixing M/N^{2} where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.

Universality of approximate message passing algorithms

We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an $n\times n$ random symmetric matrix $A$. We establish universality in noise for this AMP in the $n$-limit and validate this behavior in a number of AMPs popularly adapted in compressed sensing, statistical inferences, and optimizations in spin glasses.

Eigenvectors of Deformed Wigner Random Matrices

We investigate eigenvectors of rank-one deformations of random matrices $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^{*}$ in which $\boldsymbol A \in \mathbb R^{N \times N}$ is a Wigner real symmetric random matrix, $\theta \in \mathbb R^{+}$ , and $\boldsymbol u$ is uniformly distributed on the unit sphere. It is well known that for $\theta > 1$ the eigenvector associated with the largest eigenvalue of $\boldsymbol B$ closely estimates $\boldsymbol u$ asymptotically, while for $\theta the eigenvectors of $\boldsymbol B$ are uninformative about $\boldsymbol u$ . We examine $\mathcal O({1}/{N})$ correlation of eigenvectors with $\boldsymbol u$ before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law ${1}/{\theta ^{*} - x}$ with $\theta ^{*}:= \theta + ({1}/{\theta })$ . This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of the second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues $(-2\:,+2)$ . Therefore, most of energy of the unknown deforming vector is concentrated in a $cN$ -dimensional ( $c ) known subspace of $\boldsymbol B$ . We use a combinatorial approach to prove the result. We also extend the result to constant rank- $r$ deformations.

Local Semicircle Law Under Fourth Moment Condition

We consider a random symmetric matrix $$\mathbf{X}= [X_{jk}]_{j,k=1}^n$$ with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that $$ \max _{jk} {{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4+\delta } 0$$ , it was proved in Gotze et al. (Bernoulli 24(3):2358–2400, 2018) that with high probability the typical distance between the Stieltjes transforms $$m_n(z)$$ , $$z = u + i v$$ , of the empirical spectral distribution (ESD) and the Stieltjes transforms $$m_{\text {sc}}(z)$$ of the semicircle law is of order $$(nv)^{-1} \log n$$ . The aim of this paper is to remove $$\delta >0$$ and show that this result still holds if we assume that $$ \max _{jk} {{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4} < \infty $$ . We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.

The dimension-free structure of nonhomogeneous random matrices

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg] $$ for any $2\le p\le\infty$, where $S_p$ denotes the $p$-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case $p=\infty$, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on $\ell_2$. Along the way, we obtain optimal dimension-free bounds on the moments $(\mathbf{E}\|X\|_{S_p}^p)^{1/p}$ that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.

On the local semicircular law for Wigner ensembles

We consider a random symmetric matrix $\mathbf{X}=[X_{jk}]_{j,k=1}^{n}$ with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that $\mathbb{E}|X_{11}|^{4+\delta}=:\mu_{4+\delta} 0$. The aim of this paper is to significantly extend a recent result of the authors Gotze, Naumov and Tikhomirov (2015) and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}}\mathbf{X}$ and Wigner’s semicircle law is of order $(nv)^{-1}\log n$, where $v$ denotes the distance to the real line in the complex plane. We apply this result to the rate of convergence of the ESD to the distribution function of the semicircle law as well as to rigidity of eigenvalues and eigenvector delocalization significantly extending a recent result by Gotze, Naumov and Tikhomirov (2015). The result on delocalization is optimal by comparison with GOE ensembles. Furthermore the techniques of this paper provide a new shorter proof for the optimal $O(n^{-1})$ rate of convergence of the expected ESD to the semicircle law.

Random matrix ensembles with split limiting behavior

We introduce a new family of [Formula: see text] random real symmetric matrix ensembles, the [Formula: see text]-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but [Formula: see text] eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as [Formula: see text]; the remaining [Formula: see text] are tightly constrained near [Formula: see text] and their distribution converges to the [Formula: see text] hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We are able to isolate each regime separately through appropriate choices of weight functions for the eigenvalues and then an analysis of the resulting combinatorics.

Recovering nonuniform planted partitions via iterated projection

In the planted partition problem, the n vertices of a random graph are partitioned into k “clusters,” and edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0≤q<p≤1). We give an efficient spectral algorithm that recovers the clusters with high probability, provided that the sizes of any two clusters are either very close or separated by ≥Ω(n). We also discuss a generalization of planted partition in which the algorithm's input is not a random graph, but a random real symmetric matrix with independent above-diagonal entries.Our algorithm is an adaptation of a previous algorithm for the uniform case, i.e., when all clusters are size n/k≥Ω(n). The original algorithm recovers the clusters one by one via iterated projection: it constructs the orthogonal projection operator onto the dominant k-dimensional eigenspace of the random graph's adjacency matrix, uses it to recover one of the clusters, then deletes it and recurses on the remaining vertices. We show herein that a similar algorithm works in the nonuniform case.

A random version of principal component analysis in data clustering

Principal component analysis (PCA) is a widespread technique for data analysis that relies on the covariance/correlation matrix of the analyzed data. However, to properly work with high-dimensional data sets, PCA poses severe mathematical constraints on the minimum number of different replicates, or samples, that must be included in the analysis. Generally, improper sampling is due to a small number of data respect to the number of the degrees of freedom that characterize the ensemble. In the field of life sciences it is often important to have an algorithm that can accept poorly dimensioned data sets, including degenerated ones. Here a new random projection algorithm is proposed, in which a random symmetric matrix surrogates the covariance/correlation matrix of PCA, while maintaining the data clustering capacity. We demonstrate that what is important for clustering efficiency of PCA is not the exact form of the covariance/correlation matrix, but simply its symmetry.

Local Semicircle Law under Moment Conditions: The Stieltjes Transform, Rigidity, and Delocalization

We consider a random symmetric matrix ${X} = [X_{jk}]_{j,k=1}^n$ where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We add...