Deterministic equivalents for certain functionals of large random matrices

By Walid Hachem, Philippe Loubaton, Jamal Najim. 2005

314 citations recorded by the source corpus.

Consider a $N\times n$ random matrix $ Y_n$ where the entries are independent but not identically distributed (matrices with a variance profile) Consider now a deterministic $N\times n$ matrix $A_n$ whose columns and rows are uniformly bounded for the Euclidean norm. Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of $\Sigma_n \Sigma_n^T$ which is itself the Stieltjes transform of a probability measure. This work is motivated by the context of performance evaluation of Multiple Inputs / Multiple Output (MIMO) wireless digital communication channels. As an application, we derive a deterministic equivalent to the mutual information of a wireless channel.

Subfields: Management Information Systems, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1013056b5f234ab41c2e4300055abcdaceeeb5a2

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Citations in the corpus

Lower bounds for the smallest singular value of structured random matrices (2016, background): “The singular value distributions for structured random matrices have been studied in connection with wireless MIMO networks [13, 40].”
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On the Marčenko-Pastur law for linear time series (2013, background): “[17], who studies limiting spectral distributions of covariance matrices for data with a given variance profile.”
Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem (2013, methodology): “In the random matrix context, Problem 2 was addressed for various polynomials - and usually, also for specific choices of the distributions of the X (N) i - in work of: Marchenko-Pastur [25], Girko [19], Bai and Silverstein [13]; Tulino and Verdu [37]; Müller [27]; Moustakis and Simon [26]; Hechem,”
A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM MATRICES WITH A GIVEN VARIANCE PROFILE (2007, unknown): “f FY nY ∗ as n → ∞ and N/n → c > 0. The following theorem summarizes the ﬁrst order results needed in the sequel. It is a direct consequence of [17, Sections 2 and 4] (see also [11]): Theorem 2.3 ([17], [11]). Consider the family of random matrices (YnY∗ n ) and assume that A-1 and A-2 hold. Then”
Spectral convergence for a general class of random matrices (2011, background): “, [7,12]) that êM = êM (z) is the Stieltjes transform of a certain measure on R with total mass M (1)Tr [R] kRksup, where with the subscript sup we denote the supremum of the sequence, i.”
Capacity Analysis of One-Bit Quantized MIMO Systems With Transmitter Channel State Information (2014, background): “ is RCI 1bit = 2N r 1−H Q v u u t P t tr (HH∗)−1 . (27) As N r → +∞ and N t/N r = c (c is a constant larger than 1), the following convergence holds almost surely [42], 1 N r tr (HH∗)−1 → 1 N t −N r . (28) Therefore, we have RbCI 1bit , lim N r→+∞,N t/N r=c RCI 1bit (29) = 2N r 1−”
Capturing the learning curves of generic features maps for realistic data sets with a teacher-student model (2021, methodology): “These can be computed using classical RMT results on the concentration of the inverse of the covariance [77, 60].”
Free Probability and Random Matrices (2014, background): “These equations have actually been derived in [90] as the fixed point equations for a so-called deterministic equivalent of the square of a random matrix with noncentred, independent Gaussians with non-constant variance as entries.”
ON BILINEAR FORMS BASED ON THE RESOLVENT OF LARGE RANDOM MATRICES (2010, background): “The entries of matrices (Xn), (X n ij ; i, j, n) are complex, independent and identically distributed (i.i.d.) with mean 0 and variance 1, and An = (a n ij) is a deterministic N × n matrix whose spectral norm is bounded in n.”
Random matrix theory in statistics: A review (2014, background): “Although there are some works on Hadamard products of random matrices (Bai and Zhang, 2006; Hachem et al., 2007, 2008) which are applicable tomissing at random scenarios, a more comprehensive approach using techniques from RMT is likely to be fruitful.”
Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels Under Limited Feedback (2009, methodology): “Therefore, to be able to derive determinist ic equivalents of the SINR, we need to extend the results in [27] , [28] to account for the per-user correlation model in (7), wh ich is done in the next section.”
The Dynamics of Learning: A Random Matrix Approach (2018, background): “For certain simple distributions of M, one may define a so-called deterministic equivalent (Hachem et al., 2007; Couillet & Debbah, 2011) Q̄M for QM, which is a deterministic matrix such that for all A 2 Rn⇥n and all a,b 2 Rn of bounded (spectral and Euclidean, respectively) norms, 1 n tr (AQM) 1 n ”
Analysis of the limiting spectral measure of large random matrices of the separable covariance type (2013, background): “1 ([22], see also [12] for similar notations).”