Complex Gaussian Matrix Research Articles

Bulk universality for complex non-Hermitian matrices with independent and identically distributed entries

AbstractWe consider $$N\times N$$ N × N matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in the large N limit. Assuming the entries are independent and identically distributed (iid) with a common distribution that has finite moments, the Gaussian component is removed by the four moment theorem of Tao and Vu.

Integrated random projection and dimensionality reduction by propagating light in photonic lattices.

It is proposed that the propagation of light in disordered photonic lattices can be harnessed as a random projection that preserves distances between a set of projected vectors. This mapping is enabled by the complex evolution matrix of a photonic lattice with diagonal disorder, which turns out to be a random complex Gaussian matrix. Thus, by collecting the output light from a random subset of the waveguide channels, one can perform an embedding from a higher- to a lower-dimensional space that respects the Johnson-Lindenstrauss lemma and nearly preserves the Euclidean distances. The distance-preserving random projection through photonic lattices requires intermediate disorder levels that allow diffusive propagation of light. The proposed scheme can be utilized as a simple and powerful integrated dimension reduction stage that can greatly reduce the burden of a subsequent neural computation stage.

Complex complex landscapes

We study the saddle-points of the $p$-spin model -- the best understood example of a `complex' (rugged) landscape -- when its $N$ variables are complex. These points are the solutions to a system of $N$ random equations of degree $p-1$. We solve for $\overline{\mathcal N}$, the number of solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the B\'ezout bound $\log\overline{\mathcal N}\sim N\log(p-1)$. The Hessian of each saddle is given by a random matrix of the form $C^\dagger C$, where $C$ is a complex symmetric Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of `threshold level' well-known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the square-root of the total number of solutions are real. In terms of the complex energy, the solutions are divided into sectors where the saddles have different topological properties.

Some Characterizations of the Distribution of the Condition Number of a Complex Gaussian Matrix

AbstractThe objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices

The zeros of the random Laurent series $$1/\mu - \sum _{j=1}^\infty c_j/z^j$$ , where each $$c_j$$ is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement $$z \mapsto 1/z$$ , we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the correlation functions for the random Maclaurin series can be determined. Only for $$|\mu | \rightarrow \infty $$ is a determinantal point process obtained. For the one and two point correlations, by regarding the Maclaurin series as the limit of a random polynomial, a direct calculation can also be given.

Largest Eigenvalue Distribution of Noncircularly Symmetric Wishart-Type Matrices With Application to Hoyt-Faded MIMO Communications

This paper is concerned with the largest eigenvalue of the Wishart-type random matrix $\mathbf {{W}}=\mathbf {{X}}\mathbf {{X}}^\dagger$ (or $\mathbf {{W}}=\mathbf {{X}}^\dagger \mathbf {{X}}$ ), where $\mathbf {{X}}$ is a complex Gaussian matrix with unequal variances in the real and imaginary parts of its entries, i.e., $\mathbf {X}$ belongs to the noncircularly symmetric Gaussian subclass. By establishing a novel connection with the well-known complex Wishart ensemble, we here derive exact and asymptotic expressions for the largest eigenvalue distribution of $\mathbf {{W}}$ , which provide new insights on the effect of the real-imaginary variance imbalance of the entries of $\mathbf {X}$ . These new results are then leveraged to analyze the outage performance of multiantenna systems with maximal ratio combining subject to Nakagami- $q$ (Hoyt) fading.

On the condition of the zeros of characteristic polynomials

We prove that the expectation of the logarithm of the condition number of each of the zeros of the characteristic polynomial of a complex standard Gaussian matrix is Ω(n) (the real and imaginary parts of the entries of a Gaussian matrix are independent standard Gaussian random variables). This may provide a theoretical explanation for the common practice in numerical linear algebra that advises against computing eigenvalues via root-finding for characteristic polynomials.

Optical communication in turbid and turbulent atmosphere

Free space optical-communication (FSO) has gained significant importance due to its unique features: large bandwidth, license free spectrum, high data rate, easy and quick deployability, less power and low mass requirement. However, the performance of FSO is degraded in the turbid and turbulent atmosphere, dramatically. Various techniques are proposed to cope with the turbid media and turbulence in atmosphere, e. g. aperture averaging, diversity, adaptive optics, modulation and coding and orbital angular momentum. However, in the above systems with point-to-point optical communication structure, there exist obvious drawbacks or they are complex and expensive, and thus difficult to use in practice. In this article, array-to-point optical communication (APOC) with good performance in turbid and turbulent atmosphere is demonstrated. The strongly disturbed communication channel can be expressed as a circular complex Gaussian transmission matrix, and the transmitted field is described as a linear combination of the fields coming from different and independent segments of the digital micro-mirror device (DMD), so that the cross terms are averaged on the surface of bucket detector. Instead, the contributions of all segments for each light field nearly becomes equally weighted. Turbulence and other noises are reduced for the incoherence with sampling matrix based on the second-order correlation which has widely been used in ghost imaging and LIDAR. Consequently, narrow-band optical filter is not required at the receiver. The decoding algorithm is a new signal processing strategy from information technology, compressed sensing, which discards low frequency components in sampling process and recovers the signal by conducting convex optimization. Numerical simulations and experiments with binary and multi-bits level signals are demonstrated to show that the bit error rate of the proposed method APOC is approximately 10-4-10-2, which is feasible for the optical communication in such complex communication channels. The communication rate, limited by the frequency of the DMD and the sampling rate of the receiver, could reach hundreds of kbit/s, and with improved technology a rate of Mbit/s should be attainable. This proposed APOC could realize optical communication in turbid and turbulent atmosphere and thus offers a very effective approach to promoting the implementation in military and rescue.

Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition

The singular values squared of the random matrix product $Y = G_r G_{r-1} \cdots G_1 (G_0 + A)$, where each $G_j$ is a rectangular standard complex Gaussian matrix while $A$ is non-random, are shown to be a determinantal point process with correlation kernel given by a double contour integral. When all but finitely many eigenvalues of $A^*A$ are equal to $bN$, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of $0<b<1$ is independent of $b$, and is in fact the same as that known for the case $b=0$ due to Kuijlaars and Zhang. The critical regime of $b=1$ allows for a double scaling limit by choosing $b = (1-\tau/\sqrt{N})^{-1}$, and for this the critical kernel and outlier phenomenon are established. In the simplest case $r=0$, which is closely related to non-intersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of $b>1$ with two distinct scaling rates. Similar results also hold true for the random matrix product $T_r T_{r-1} \cdots T_1 (G_0 + A)$, with each $T_j$ being a truncated unitary matrix.

Dynamical correlation functions for products of random matrices

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard–Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.

Lyapunov Exponents for Products of Complex Gaussian Random Matrices

The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = \Sigma^{1/2} G_i^{\rm c}$, where $\Sigma$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.

Almost Sure Localization of the Eigenvalues in a Gaussian Information Plus Noise Model. Application to the Spiked Models.

Let $S$ be a $M$ times $N$ random matrix defined by $S = B + \sigma W$ where $B$ is a uniformly bounded deterministic matrix and where $W$ is an independent identically distributed complex Gaussian matrix with zero mean and variance $1/N$ entries. The purpose of this paper is to study the almost sure location of the eigenvalues of the Gram matrix $SS^*$ when $M$ and $N$ converge to infinity such that the ratio $M/N$ converges towards a constant $c &gt; 0$. The results are used in order to derive, using an alternative approach, known results concerning the behavior of the largest eigenvalues of $SS^*$ when the rank of $B$ remains fixed and $M$ and $N$ converge to infinity.

Block diagonalization for multi-user MIMO with other-cell interference

Block diagonalization is one approach for linear preceding in the multiple-input multiple-output broadcast channel that sends multiple interference free data streams to different users in the same cell. Unfortunately, block diagonalization neglects other-cell interference (OCI), which limits the performance of users at the edge of the cell. This paper presents an OCI-aware enhancement to block diagonalization that uses a whitening filter for interference suppression at the receiver and a novel precoder using the interference-plus-noise covariance matrix for each user at the transmitter. For complex Gaussian matrix channels, the asymptotic sum rate of the proposed system is analyzed under a large antenna assumption for isotropic inputs and compared to conventional block diagonalization. The capacity loss due to OCI is quantified in terms of results from single-user MIMO capacity. Several numerical examples compare achievable sum rates, the proposed asymptotic rates, and the capacity loss, in low and high interference regimes.

Eigenvalue distributions for some correlated complex sample covariance matrices

The distributions of the smallest and largest eigenvalues for the matrix product Z†Z, where Z is an n × m complex Gaussian matrix with correlations both along rows and down columns, are expressed as m × m determinants. In the case of correlation along rows, these expressions are computationally more efficient for the purpose of tabulation than those involving sums over partitions and Schur polynomials reported recently for the same distributions.

Distribution and characteristic functions for correlated complex Wishart matrices

Let A ( t ) be a complex Wishart process defined in terms of the M × N complex Gaussian matrix X ( t ) by A ( t ) = X ( t ) X ( t ) H . The covariance matrix of the columns of X ( t ) is Σ . If X ( t ) , the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t 1 , t 2 , where t 1 < t 2 . In particular, we derive the following results: the joint density of the elements of A ( t 1 ) , A ( t 2 ) , the joint density of the eigenvalues of Σ - 1 A ( t 1 ) , Σ - 1 A ( t 2 ) , the characteristic function of the elements of A ( t 1 ) , A ( t 2 ) , the characteristic function of the eigenvalues of Σ - 1 A ( t 1 ) , Σ - 1 A ( t 2 ) . In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.

Asymptotic Analysis in MIMO MRT/MRC Systems

Through the analysis of the probability density function of the squared largest singular value of a complex Gaussian matrix at the origin and tail, we obtain two asymptotic results related to the multi-input multi-output (MIMO) maximum-ratio-transmission/maximum-ratio-combining (MRT/MRC) systems. One is the asymptotic error performance (in terms of SNR) in a single-user system, and the other is the asymptotic system capacity (in terms of the number of users) in the multiuser scenario when multiuser diversity is exploited. Similar results are also obtained for two other MIMO diversity schemes, space-time block coding and selection combining. Our results reveal a simple connection with system parameters, providing good insights for the design of MIMO diversity systems.

Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices

AbstractWe compute the limiting distributions of the largest eigenvalue of a complex Gaussian samplecovariance matrix when both the number of samples and the number of variables in each samplebecome large. When all but finitely many, say r, eigenvalues of the covariance matrix arethe same, the dependence of the limiting distribution of the largest eigenvalue of the samplecovariance matrix on those distinguished r eigenvalues of the covariance matrix is completelycharacterized in terms of an infinite sequence of new distribution functions that generalizethe Tracy-Widom distributions of the random matrix theory. Especially a phase transitionphenomena is observed. Our results also apply to a last passage percolation model and aqueuing model. 1 Introduction Consider M independent, identically distributed samples y 1 ,...,~y M , all of which are N ×1 columnvectors. We further assume that the sample vectors ~y k are Gaussian with mean µ and covarianceΣ, where Σ is a fixed N ×N positive matrix; the density of a sample ~y isp(~y) =1(2π)

A Determinant Representation for the Distribution of a Generalised Quadratic Form in Complex Normal Vectors

Consider the quadratic form Z=YH(XLXH)−1Y where Y is a p×m complex Gaussian matrix, X is an independent p×n complex Gaussian matrix, L is a Hermitian positive definite matrix, and m⩽p⩽n. The distribution of Z has been studied for over 30 years due to its importance in certain multivariate statistics but no satisfactory numerical methods for computing this distribution appear to be available. Hence this paper deals with a representation for the density function of Z in terms of a ratio of determinants which is shown to be more amenable to numerical work than previous representations, at least for small values of p. Also for m=1 this work has applications in digital mobile radio for a specific channel where p antennas are used to receive a signal with n interferers. Some of these applications in radio communication systems are discussed.

On the spectra of Gaussian matrices

We give a simple characterization of the moduli of the eigenvalues of a complex Gaussian matrix in terms of x2 distributions. We also show that the spectral radius of a k × k complex Gaussian matrix is stochastically smaller than the norm of a k × (k + 1) real Gaussian matrix.