Asymptotically liberating sequences of random unitary matrices

Anderson, Greg W.; Farrell, Brendan

doi:10.1016/j.aim.2013.12.026

Cited by 31 publications

(32 citation statements)

References 27 publications

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“…It is a challenging issue in random matrix theory to weaken the right-orthogonal invariance to more practical assumptions, including discrete cosine transform (DCT) or Hadamard matrices with random permutation [53]. See [54], [55] for a theoretical progress in this direction.…”

Section: A State Evolutionmentioning

confidence: 99%

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On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy

Takeuchi¹

2021

Preprint

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Orthogonal/vector approximate message-passing (AMP) is a powerful message-passing (MP) algorithm for signal reconstruction in compressed sensing. This paper proves the convergence of Bayes-optimal orthogonal/vector AMP in the large system limit. The proof strategy is based on a novel long-memory (LM) MP approach: A first step is a construction of LM-MP that is guaranteed to converge in principle. A second step is a large-system analysis of LM-MP via an existing framework of state evolution. A third step is to prove the convergence of Bayes-optimal LM-MP via a new statistical interpretation of existing LM damping. The last is an exact reduction of Bayes-optimal LM-MP to Bayes-optimal orthogonal/vector AMP. Numerical simulations are presented to show the verification of state evolution results for damped orthogonal/vector AMP and a challenge in LM-MP for finite-sized systems.

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Section: A State Evolutionmentioning

confidence: 99%

“…To reduce the computational complexity, we assume the SVD structure A = ΣV T , in which V denotes a Hadamard matrix with random row permutation. Such orthogonal matrices V may be regarded as a practical alternative of Haar orthogonal matrices [54], [55].…”

Section: A Simulation Conditionsmentioning

confidence: 99%

On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy

Takeuchi¹

2021

Preprint

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“…spectral distribution [23,17,22,1,10]; asymptotic non-commutative independence between random matrix ensembles [34,35,2,25], and central limit theorems for linear statistics [12,19,4,3,31].…”

Section: Introductionmentioning

confidence: 99%

On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Fréchet Variation

Dı́az¹,

Mingo²

2022

Preprint

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In this paper we propose a new approach to the central limit theorem (CLT), based on functions of bounded Féchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on: a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and, as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g., the CLT for the continuously differentiable linear statistics of block Gaussian matrices.In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, G 2 . Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in G 2 .

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“…where A N,k denotes the matrix of trace zero p k (D N,i k ) − tr(p k (D N,i k ))I N , but (1.1) also implies that sup N A N,k < ∞, and hence (AF.2) holds. As it was intended, independent Haar-unitary random matrix ensembles and independent Haar-orthogonal random matrix ensembles are among those unitary random matrix ensembles shown to be asymptotically liberating, see Theorem 2.8 in [1] or Lemma 6 below.…”

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confidence: 98%

“…Aiming to enclose all of those unitary random matrix ensembles that give rise to asymptotic free independence when used for conjugation, B. Farrell and G. Anderson introduced in [1] the notion of asymptotically liberating random matrix ensembles. Definition 2.…”

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confidence: 99%

Fluctuation moments induced by conjugation with asymptotically liberating random matrix ensembles

Vazquez-Becerra¹

2020

Preprint

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Anderson and B. Farrel showed that conjugation of constant matrices by asymptotically liberating random unitary matrices give rise to asymptotic free independence. Independent Haar-unitary random matrices and independent Haar-orthogonal random matrices are examples of asymptotically liberating ensembles. In this paper, we investigate the fluctuation moments, and higher order moments, induced on constant matrices by conjugation with asymptotically liberating ensembles. In particular, we determine fluctuation moments induced by ensembles related to the Discrete Fourier Transform matrix.

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Asymptotically liberating sequences of random unitary matrices

Cited by 31 publications

References 27 publications

On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy

On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy

On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Fréchet Variation

Fluctuation moments induced by conjugation with asymptotically liberating random matrix ensembles

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