Brian Rider

Quantization Bounds on Grassmann Manifolds and Applications to MIMO Communications

The Grassmann manifold Gn,p (L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space Ln, where L is either R or C. This paper considers the quantization problem in which a source in Gn,p (L) is quantized through a code in Gn,q (L), with p and q not necessarily the same. The analysis is based on the volume of a metric ball in Gn,p (L) with center in Gn,q (L), and our chief result is a closed-form expression for the volume of a metric ball of radius at most one. This volume formula holds for arbitrary n, p, q, and L, while previous results pertained only to some special cases. Based on this volume formula, several bounds are derived for the rate-distortion tradeoff assuming that the quantization rate is sufficiently high. The lower and upper bounds on the distortion rate function are asymptotically identical, and therefore precisely quantify the asymptotic rate-distortion tradeoff. We also show that random codes are asymptotically optimal in the sense that they achieve the minimum possible distortion in probability as n and the code rate approach infinity linearly. Finally, as an application of the derived results to communication theory, we quantify the effect of beamforming matrix selection in multiple-antenna communication systems with finite rate channel state feedback.

On the Information Rate of MIMO Systems With Finite Rate Channel State Feedback Using Beamforming and Power On/Off Strategy

It is well known that multiple-input multiple-output (MIMO) systems have high spectral efficiency, especially when channel state information at the transmitter (CSIT) is available. In many practical systems, it is reasonable to assume that the CSIT is obtained by a limited (i.e., finite rate) feedback and is therefore imperfect. We consider the design problem of how to use the limited feedback resource to maximize the achievable information rate. In particular, we develop a low complexity power on/off strategy with beamforming (or Grassmann precoding), and analytically characterize its performance. Given the eigenvalue decomposition of the covariance matrix of the transmitted signal, refer to the eigenvectors as beams, and to the corresponding eigenvalues as the beams power. A power on/off strategy means that a beam is either turned on with a constant power, or turned off. We will first assume that the beams match the channel perfectly and show that the ratio between the optimal number of beams turned on and the number of antennas converges to a constant when the numbers of transmit and receive antennas approach infinity proportionally. This motivates our power on/off strategy where the number of beams turned on is independent of channel realizations but is a function of the signal-to-noise ratio (SNR). When the feedback rate is finite, beamforming cannot be perfect, and we characterize the effect of imperfect beamforming by quantization bounds on the Grassmann manifold. By combining the results for power on/off and beamforming, a good approximation to the achievable information rate is derived. Simulations show that the proposed strategy is near optimal and the performance approximation is accurate for all experimented SNRs.

A limit theorem at the edge of a non-Hermitian random matrix ensemble

The study of the edge behaviour in the classical ensembles of Gaussian Hermitian matrices has led to the celebrated distributions of Tracy–Widom. Here we take up a similar line of inquiry in the non-Hermitian setting. We focus on the family of N × N random matrices with all entries independent and distributed as complex Gaussian of mean zero and variance 1/N. This is a fundamental non-Hermitian ensemble for which the eigenvalue density is known. Using this density, our main result is a limit law for the (scaled) spectral radius as N ↑ ∞. As a corollary, we get the analogous statement for the case where the complex Gaussians are replaced by quaternion Gaussians.

Diffusion at the Random Matrix Hard Edge

We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian sample-covariance structures (beta ensembles) are described by the spectrum of a random diffusion generator. This generator may be mapped onto the “Stochastic Bessel Operator,” introduced and studied by A. Edelman and B. Sutton in [6] where the corresponding convergence was first conjectured. Here, by a Riccati transformation, we also obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. All this pertains to the so-called hard edge of random matrix theory and sits in complement to the recent work [15] of the authors and B. Virág on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used below to prove there exists a transition between the soft and hard edge laws at all values of beta.

Concentration of permanent estimators for certain large matrices

Let An = (aij) n=1 be an n n positive matrix with entries in [a;b]; 0 < a b. Let Xn = ( p aijxij) n=1 be a random matrix wherefxijg are i.i.d. N(0; 1) random variables. We show that for large n, det(X T n Xn) concentrates sharply at the permanent of An, in the sense that n 1 log(det(X T n Xn)=per An)!n!1 0 in probability.

Universality of the Stochastic Airy Operator

We introduce a new method for studying universality of random matrices. Let T-n be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, Tn converges to the stochastic Airy operator. In particular, the top edge of the Dyson beta ensemble and the corresponding eigenvectors are universal. As a byproduct, these ideas lead to conjectured operator limits for the entire family of soft edge distributions

Gaussian fluctuations for non-Hermitian random matrix ensembles

Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of

Extremal laws for the real Ginibre ensemble

1/\sqrt{N}

On the information rate of MIMO systems with finite rate channel state feedback and power on/off strategy

and letting N→∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type XN(f)=∑k=1Nf(λk) where λ1, λ2, …, λN denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533–605], where the analogous result for random sample covariance matrices is established.

Order Statistics and Ginibre's Ensembles

The real Ginibre ensemble refers to the family of n n matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as n!1. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.