Methods for Generating Random Orthogonal Matrices

Some well-known real-number codes are DFT codes. Since these codes are cyclic, they can be used to correct erasures (errors at known positions) and detect errors, using the locator polynomial via the syndrome, with efficient algorithms. The stability of such codes are, however, very poor for burst error patterns. In such conditions, the stability of the system of equations to be solved is very poor. This amplifies the rounding errors inherent to the real number field. In order to improve the stability of real-number error-correcting codes, other types of coding matrices were considered, namely random orthogonal matrices. These type of codes have proven to be very stable, when compared to DFT codes. However, the problem of detecting errors (when the positions of these errors are not known) with random codes was not addressed. Such codes do not possess any specific structure which could be exploited to create an efficient algorithm. In this paper, we present an efficient method to locate errors with codes based on random orthogonal matrices. Index Terms — Error correction, random matrices, real number codes, sparse solutions

ABSTRACTA truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product Pm of m independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of Pm form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulas, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity.

The quantum chaos conjecture applied to a finite dimensional quantum system implies that such a system has eigenstates that show similar statistical properties as the column vectors of random orthogonal or unitary matrices. Here, we consider the different probabilities for obtaining a specific outcome in a projective measurement, provided the system is in one of its eigenstates. We then give analytic expressions for the joint probability density for these probabilities, with respect to the ensemble of random matrices. In the case of the unitary group, our results can be applied, also, to the phenomenon of universal conductance fluctuations, where the same mathematical quantities describe partial conductances in a two-terminal mesoscopic scattering problem with a finite number of modes in each terminal.

We study the ability of linear recurrent networks obeying discrete time dynamics to store long temporal sequences that are retrievable from the instantaneous state of the network. We calculate this temporal memory capacity for both distributed shift register and random orthogonal connectivity matrices. We show that the memory capacity of these networks scales with system size.

In the study of random matrices, Lyapunov exponents characterize the rate of exponential growth of the singular values of product of increasing number of |$i.i.d$| random matrices of fixed order. These have been well studied objects in the literature. Only recently, absolute values of the eigenvalues of such products of random matrices have come under analytical study. The quantities that characterize the rate of exponential growth of the absolute values of eigenvalues have been named stability exponents, in order to distinguish them from that of the singular values. In the cases of Ginibre matrices and truncated Haar unitary matrices, the stability exponents have been observed to match with the Lyapunov exponents. In this article, we generalize this result to the case of all isotropic random matrices. We also derive the asymptotic joint probability distributions for the fluctuations of both the singular values and the absolute values of eigenvalues of product of increasing number of |$i.i.d$| real or complex isotropic random matrices of fixed order. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of this, we show the probability that the product of |$n$||$i.i.d$| real isotropic random matrices has all the eigenvalues real goes to one as |$n \to \infty$|⁠.

In this paper we apply the REFORM method[3, 4] to the deduction of the system of canonical equations for normalized spectral functions of the matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is random matrix from class C 12 [26] or from the following class C 13 of distributions of random matrices: where are independent random real matrices whose entries are independent for every n, and for a certain δ > 0 the Lyapunov condition is fulfilled: This problem has been considered in some publications for matrices A n + U n , where Unitary matrix U n from the class C 1 [26] is distributed by Haar measure, at the “ad hoc” level, without a strong proof, on the basis of heuristic calculations. Therefore, the behavior of limit n.s.f. of the sum of a random Unitary matrix U n and a nonrandom matrix A n has not been discovered. Many conclusions of various kinds have been presented in the literature (see [13–27]) for random matrices A n + Ξ n B n , where Ξ n is the random matrix with independent entries, concerning, for example, the effectiveness of the REFORM method and the role of the martingale difference representation for the resolvent of random matrices. We give the formulation of the Circular Law for random matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is from the so called Class C 12 of random Unitary matrices (see[26]). In this case the Circular law means that the support of accompanying spectral density p ( x, y ) of eigenvalues of A n + B n U n ( ε ) C n looks like several drops of mercury on a table, and inside of some drops it is possible that some dry circles appear. We call this support of limit density p ( x, y ) the Mercury support . If the distances between the centers of these drops are large enough we have several separate almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices [28, 29]. According to the distances between the centers of these drops, these drops might not touch each other or can merge creating fanciful shapes represented at the end of the paper, Figures 1–16. The analogy with the Circular Law is the following: for a simple Unitary matrix U n from class C 1 all its eigenvalues are distributed asymptotically uniformly on the circumference and, evidently, this limit distribution does not coincide with the uniform distribution on the circle. But if we will add to Unitary matrix U n any diagonal nonrandom matrix A n = ( δ ij a j ) then if the distances between diagonal entries a j are large enough we have almost the same picture of limit distribution of eigenvalues of U n + A n if Unitary matrix U n would be equal to the random matrix Ξ n with random independent entries satisfying the Global Circular Law (see Part I of this paper). The rough explanation of this phenomenon is the following: the expected scalar products of the vector row or vector column of matrix Ξ n have the order n −1 . Therefore, the matrix Ξ n looks like orthogonal random matrix. In this paper we use for matrices A n + B n U n ( ε ) C n the triply regularized V -transform[2, 5],( VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Randomly Independent Arrays ) where α > 0, ε ≠= 0, γ ≠= 0 are regularization parameters , τ = t + is is a complex number and µ n { x,G n ( t, s, γ, ε )} is the normalized spectral function of G -matrix: Such a V -transformation was used for the first time in 1982–1985 in [2,3,5]. Canonical equations for the limit distribution function of normalized spectral functions of some matrices A n + B n U n and the following V - domain G δ , δ > 0(see [27, pp.142, 173]) of the pseudodistribution of eigenvalues of matrices A n + B n U n are found: where are eigenvalues of the matrix.

A new factorization of orthogonal matrices is proposed that is based on Givens-Jacobi rotations but not on the QR decomposition. Rotations are arranged more uniformly than in the known factorizations that use them, so that more rotations can be computed in parallel, and fewer layers of concurrent rotations are necessary to model a matrix. Therefore, throughput can be increased, and latency can be reduced, compared to the known solutions, even though the obtainable gains highly depend on application specificity, software-hardware architecture and matrix size. The proposed approach allows for developing more efficient algorithms and hardware for generating random matrices, for optimizing matrices, and for processing data with linear transformations. We have verified this by implementing and evaluating a multithreaded Java application for generating random orthogonal matrices.

We consider powers of the absolute value of the characteristic polynomial of Haar distributed random orthogonal or symplectic matrices, as well as powers of the exponential of its argument, as a random measure on the unit circle. We also consider the case where these measures are restricted to the unit circle minus small neighborhoods around ±1. We show that for small enough powers and under suitable normalization, as the matrix size goes to infinity, these random measures converge in distribution to a Gaussian multiplicative chaos (GMC) measure. Our result is analogous to one relating to unitary matrices previously established by Christian Webb (2015 Electron. J. Probab. 20). We thus complete the connection between the classical compact groups and GMC. To prove this convergence when excluding small neighborhoods around ±1 we establish appropriate asymptotic formulae for Toeplitz and Toeplitz + Hankel determinants with merging singularities. Using a recent formula due to Claeys et al (2021 Int. Math. Res. Not. rnaa354), we are able to prove convergence on the whole of the unit circle.

We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [J. Pennington, S. Schoenholz and S. Ganguli, The emergence of spectral universality in deep networks, Proc. Mach. Learn. Res. 84 (2018) 1924–1932, arXiv:1802.09979] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability in this case has to be justified. The justification has been given in [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper, we use another version of the techniques to extend the results of [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] to the case where the entries of the data matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, justifies the mean field approximation in the infinite width limit for the deep untrained neural networks and the property of the macroscopic universality of random matrix theory in this case.

We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d=2 (finitely connected XY spins with random chiral interactions) and for d=3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

Vector Symbolic Architectures (VSA) are methods designed to enable distributed representation and manipulation of semantically-structured information, such as natural languages. Recently, a new VSA based on multiplication of distributed vectors by random matrices was proposed, this is known as Matrix-Binding-of-Additive-Terms (MBAT). We propose an enhancement that introduces an important additional feature to MBAT: the ability to 'unbind' symbols. We show that our method, which exploits the inherent properties of orthogonal matrices, imparts MBAT with the 'question answering' ability found in other VSAs. We compare our results with another popular VSA that was recently demonstrated to have high utility in brain-inspired machine learning applications.

ROM Simulation with Random Rotation Matrices

On considere le produit Ψn de n matrices aleatoires M i.i.d. ayant les proprietes suivantes: la suite (Ψn)n est bornee et M possede un vecteur invariant deterministe (constant). On suppose que la probabilite pour que, sur le cercle unite, M possede une unique valeur propre simple en 1 soit non nulle. On montre que Ψn est la somme d’un processus fluctuant et d’un processus decroissant. Ce dernier tend vers zero presque surement et exponentiellement rapidement lorsque n tend vers l’infini. Le terme fluctuant converge en moyenne de Cesaro vers une limite caracterisee explicitement par le vecteur invariant deterministe et par les donnees spectrales associees a la valeur propre 1 de la matrice $\mathbb{E}[M]$. Aucune hypothese supplementaire n’est faite sur les matrices M; elles peuvent etre a valeurs complexes et pas necessairement inversibles. On applique les resultats generaux a deux classes de systemes dynamiques: les chaines de Markov inhomogenes avec matrices de transition aleatoires (matrices stochastiques), ainsi que les systemes quantiques avec interactions repetees et aleatoires. Dans les deux cas, on prouve des resultats d’ergodicite pour la dynamique et on determine les etats limites.

Products and sums of random matrices have seen a rapid development in the\npast decade due to various analytical techniques available. Two of these are\nthe harmonic analysis approach and the concept of polynomial ensembles. Very\nrecently, it has been shown for products of real matrices with anti-symmetric\nmatrices of even dimension that the traditional harmonic analysis on matrix\ngroups developed by Harish-Chandra et al. needs to be modified when considering\nthe group action on general symmetric spaces of matrices. In the present work,\nwe consider the product of complex random matrices with Hermitian matrices, in\nparticular the former can be also rectangular while the latter has not to be\npositive definite and is considered as a fixed matrix as well as a random\nmatrix. This generalises an approach for products involving the Gaussian\nunitary ensemble (GUE) and circumvents the use there of non-compact group\nintegrals. We derive the joint probability density function of the real\neigenvalues and, additionally, prove transformation formulas for the\nbi-orthogonal functions and kernels.\n

The probability that there are k real eigenvalues for an n-dimensional real random matrix is known. Here, we study this for the case of products of independent random matrices. Relating the problem of the probability that the product of two real two-dimensional random matrices has real eigenvalues to an issue of optimal quantum entanglement, this is fully analytically solved. It is shown that in π/4 fraction of such products the eigenvalues are real. Being greater than the corresponding known probability () for a single matrix, it is shown numerically that the probability that all eigenvalues of a product of random matrices are real tends to unity as the number of matrices in the product increases indefinitely. Some other numerical explorations, including the expected number of real eigenvalues, are also presented, where an exponential approach of the expected number to the dimension of the matrix seems to hold.