Random Gram Research Articles

On the fourth moment of a random determinant

In this paper, we generalize the formula for the fourth moment of a random determinant to account for entries with asymmetric distribution. We also derive the second moment of a random Gram determinant.

Some random Fourier matrices and their singular values

This paper is centered on the spectral study of Random Fourier matrices, that is, n′×n matrices A whose (j,k) entries are exp⁡(2iπm〈Xj,Yk〉), with two independent sequences, Xj and Yk, of random vectors, and where m is a real number that measures the sizes of oscillations. We are interested in large values of m, that is, high frequencies. This is a companion paper of [3], which is centered on random variables uniformly distributed on a symmetric interval. We refer to its bibliography for the wide literature on random Gram matrices, which are involved in this study. We show here that its results may be generalized in different directions. Some of them are straightforward, such as the comparison between spectra of random Fourier matrices and spectra of corresponding integral operators. We concentrate here on asymptotic formulas for Hilbert-Schmidt norms, which allows us to generalize estimates related to the number of degrees of freedom, which had been previously given for the sinc kernel.

A graphon approach to limiting spectral distributions of Wigner‐type matrices

We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity ω(1/n): inhomogeneous random graphs with roughly equal expected degrees, W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.

Coulomb Gas Analogy: A Statistical Physics Approach to Performance Analysis of MIMO Systems

In this correspondence, by adopting the Coulomb gas analogy that is an analytical approach from statistical physics, we derive a closedform approximation of the cumulative distribution function (CDF) of the largest eigenvalue of a random Gram matrix H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sup> H, where H denotes the channel matrix of a multiple-input multiple-output (MIMO) system. The expression of approximation has a simple structure in terms of several elementary functions, which is derived by assuming that the entries of H have Nakagami-m distributed amplitude with an independent phase. This assumption is made because it applies in two common cases of the Nakagami-m distributions, i.e., with a uniformly distributed phase and a complex signal model. Simulation results indicate that the theoretical expression also provides a good approximation to the CDF for Rice and Hoyt distributions when the CDF is within a near-one range. This range takes the form of (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">no</sub> ,1), where we assume P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">no</sub> = 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> by default in this study. The derived approximation can find many uses for the performance analysis of MIMO beamforming and singular value decomposition MIMO systems, e.g., evaluating the outage probability for these systems.

Numerically Stable Evaluation of Moments of Random Gram Matrices With Applications

This paper is focuses on the computation of the positive moments of one-side correlated random Gram matrices. Closed-form expressions for the moments can be obtained easily, but numerical evaluation thereof is prone to numerical stability, especially in high-dimensional settings. This letter provides a numerically stable method that efficiently computes the positive moments in closed-form. The developed expressions are more accurate and can lead to higher accuracy levels when fed to moment based-approaches. As an application, we show how the obtained moments can be used to approximate the marginal distribution of the eigenvalues of random Gram matrices.

Local law for random Gram matrices

We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of $XX^*$.

Singularities of the density of states of random Gram matrices

For large random matrices $X$ with independent, centered entries but not necessarily identical variances, the eigenvalue density of $XX^*$ is well-approximated by a deterministic measure on $\mathbb{R} $. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.

Analytical Derivation of the Inverse Moments of One-Sided Correlated Gram Matrices With Applications

This paper addresses the development of analytical tools for the computation of the moments of random Gram matrices with one side correlation. Such a question is mainly driven by applications in signal processing and wireless communications wherein such matrices naturally arise. In particular, we derive closed-form expressions for the inverse moments and show that the obtained results can help approximate several performance metrics such as the average estimation error corresponding to the Best Linear Unbiased Estimator (BLUE) and the Linear Minimum Mean Square Error LMMSE or also other loss functions used to measure the accuracy of covariance matrix estimates.

The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.

Spectra of infinite-dimensional sample covariance matrices

We study spectral functions of infinite-dimensional random Gram matrices of the form RRT, where R is a rectangular matrix with an infinite number of rows and with the number of columns N → ∞, and the spectral functions of infinite sample covariance matrices calculated for samples of volume N → ∞ under conditions analogous to the Kolmogorov asymptotic conditions. We assume that the traces d of the expectations of these matrices increase with the number N such that the ratio d/N tends to a constant. We find the limiting nonlinear equations relating the spectral functions of random and nonrandom matrices and establish the asymptotic expression for the resolvent of random matrices.

Stochastic canonical equation for normalized spectral functions of random Gram matrices

Article Stochastic canonical equation for normalized spectral functions of random Gram matrices was published on January 1, 2000 in the journal Random Operators and Stochastic Equations (volume 8, issue 4).