Operator-valued Free Probability Theory Research Articles

Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory

AbstractWe address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.

Berry-Esseen bounds for the multivariate ℬ-free CLT and operator-valued matrices

We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued free probability theory such as the operator-valued free Central Limit Theorem and the asymptotic behaviour of distributions of operator-valued matrices. Our estimates are on the level of operator-valued Cauchy transforms and the Lévy distance. We address the single-variable as well as the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the Lévy distance for the operator-valued free Central Limit Theorem. Our results also yield quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. In the scalar-valued multivariate case, these estimates could be passed to explicit bounds on the order of convergence under the Kolmogorov distance.

Channel Estimation and Robust Detection for IQ Imbalanced Uplink Massive MIMO-OFDM With Adjustable Phase Shift Pilots

In this paper, we investigate channel estimation and robust detection for uplink massive multi-input multi-output orthogonal frequency division multiplexing (MIMO-OFDM) systems with in-phase and quadrature-phase imbalances (IQI). We represent the received signal with a real-valued model, where the combination of the IQI and the wireless channel is treated as the effective channel. Based on the real-valued model, we then perform the minimum mean square error (MMSE) criterion based estimation for the effective channel. We use adjustable phase shift pilots (APSPs) to reduce the pilot overhead and verify that the MSE of the effective channel estimation can be minimized by scheduling the pilot phase shifts properly. We propose a pilot phase shift scheduling algorithm based on the covariance matrices of the effective channels. Since the channel estimation performance might be degraded due to using APSPs, we develop an MMSE criterion based data detection scheme which is robust to the channel estimation error. Using operator-valued free probability theory, we derive an asymptotic deterministic equivalent for the ergodic achievable sum rate of the proposed detection scheme. The performance of the channel estimation with APSPs and the robust data detection is demonstrated via extensive numerical results.

Widely-Linear Processing for the Uplink of the Massive MIMO With IQ Imbalance: Channel Estimation and Data Detection

In this paper, we investigate widely-linear channel estimation and data detection for the uplink of the massive multiple-input multiple-output (MIMO) with in-phase or quadrature-phase imbalance (IQI). We firstly investigate widely-linear minimum mean square error (WL-MMSE) estimation for the effective channel which is the product of the IQI coefficients and the wireless channel. We reuse pilots among different user terminals (UTs) to reduce pilot overhead and derive a lower bound of the estimation MSE. Based on the optimal condition under which the lower bound is achieved, we show that if we assign the same pilot for the UTs of which the covariance matrices are orthogonal, the pilot interference does not affect the MSE performance. Then according to the obtained estimates of the effective channel, we further propose WL-MMSE data detection that is robust to the estimation error. The proposed widely-linear approach can cope with the IQI and multi-user interference jointly. With some results of operator-valued free probability theory, we derive an asymptotic deterministic equivalent for the ergodic achievable sum rate of the proposed robust WL-MMSE data detection. The performance of the proposed widely-linear processing approach is verified by the numerical results.

EIGENVALUE DISTRIBUTIONS OF VARIANCE COMPONENTS ESTIMATORS IN HIGH-DIMENSIONAL RANDOM EFFECTS MODELS.

We study the spectra of MANOVA estimators for variance component covariance matrices in multivariate random effects models. When the dimensionality of the observations is large and comparable to the number of realizations of each random effect, we show that the empirical spectra of such estimators are well-approximated by deterministic laws. The Stieltjes transforms of these laws are characterized by systems of fixed-point equations, which are numerically solvable by a simple iterative procedure. Our proof uses operator-valued free probability theory, and we establish a general asymptotic freeness result for families of rectangular orthogonally-invariant random matrices, which is of independent interest. Our work is motivated in part by the estimation of components of covariance between multiple phenotypic traits in quantitative genetics, and we specialize our results to common experimental designs that arise in this application.

English

An operadic framework is developed to explain the inversion formula relating moments and cumulants in operator-valued free probability theory.

Free Deterministic Equivalents for the Analysis of MIMO Multiple Access Channel

In this paper, a free deterministic equivalent is proposed for the capacity analysis of the multi-input multi-output (MIMO) multiple access channel (MAC) with a more general channel model compared to previous works. In particular, a MIMO MAC with one base station (BS) equipped with several distributed antenna sets is considered. Each link between a user and a BS antenna set forms a jointly correlated Rician fading channel. The analysis is based on operator-valued free probability theory, which broadens the range of applicability of free probability techniques tremendously. By replacing independent Gaussian random matrices with operator-valued random variables satisfying certain operator-valued freeness relations, the free deterministic equivalent of the considered channel Gram matrix is obtained. The Shannon transform of the free deterministic equivalent is derived, which provides an approximate expression for the ergodic input-output mutual information of the channel. The sum-rate capacity achieving input covariance matrices are also derived based on the approximate ergodic input-output mutual information. The free deterministic equivalent results are easy to compute, and simulation results show that these approximations are numerically accurate and computationally efficient.

On the asymptotic distribution of block-modified random matrices

We study random matrices acting on tensor product spaces which have been transformed by a linear block operation. Using operator-valued free probability theory, under some mild assumptions on the linear map acting on the blocks, we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution. Moreover, using recent results on operator-valued subordination, we present an algorithm that computes, numerically but in full generality, the limiting eigenvalue distribution of the modified matrices. Our analytical results cover many cases of interest in quantum information theory: we unify some known results and we obtain new distributions and various generalizations.

Analytic function theory for operator-valued free probability

Abstract It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a main object of study in non-commutative probability theory as the function theory encodes information on the probability measures and the various convolution operations. In extending this theory to operator-valued free probability theory, the analogue of the Cauchy transform is a non-commutative function with domain equal to the non-commutative upper-half plane. In this paper, we prove an analogous characterization of the Cauchy transforms, again, entirely in terms of their analytic and asymptotic behavior. We further characterize those functions which arise as the Voiculescu transform of ⊞ {\boxplus} -infinitely divisible operator-valued distributions. As consequences of these results, we provide a characterization of infinite divisibility in terms of the domain of the relevant Voiculescu transform, provide a purely analytic definition of the semigroups of completely positive maps associated to infinitely divisible distributions and provide a Nevanlinna representation for non-commutative functions with the appropriate asymptotic behavior.

Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

We show that the quadratic matrix equation VW + η(W)W = I, for given V with positive real part and given analytic mapping η with some positivity preserving properties, has exactly one solution W with positive real part. Also we provide and compare numerical algorithms based on the iteration underlying our proofs. This work bears on operator-valued free probability theory, in particular on the determination of the asymptotic eigenvalue distribution of band or block random matrices.

Gaussian random band matrices and operator-valued free probability theory

1. Gaussian random band matrices. Let g ij , 1 ≤ i, j ≤ n, k = 1, . . . , r be independent centered complex Gaussian random variables of covariance 1 n (1 + δij) (i.e., the expectation E(g ijg k ij) = 1 n (1 + δij)), such that g k ij = g k ji. Let for each i, j, k, σn(i, j; k) be a positive real number, such that σn(i, j; k) = σn(j, i; k). Consider the r-tuple of n × n matrices Gn(k) = (σn(i, j; k)g k ij)ij , k = 1, . . . , r. The family {Gn(k)}k is called a family of independent Gaussian random band matrices. As an example, assume that σn(i, j; k) = 1 if |i − j| < cn, and is zero otherwise. In this case all entries of Gn(k) are zero outside a band around the diagonal; this is the reason such matrices are called band matrices. In the case that σn(i, j; k) is identically 1 one recovers the so-called Gaussian Unitary Ensemble, see e.g. [9].

Combinatorial theory of the free product with amalgamation and operator-valued free probability theory

Preliminaries on non-crossing partitions Operator-valued multiplicative functions on the lattice of non-crossing partitions Amalgamated free products Operator-valued free probability theory Operator-valued stochastic processes and stochastic differential equations Bibliography.