Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and free probability theory

Tanaka, Toshiki

doi:10.1088/1742-6596/95/1/012002

Cited by 12 publications

(11 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(B.4) is the Orthogonal low-rank version of the Harish-Chandra-Itzykson-Zuber integrals ( [44], [45]). The result is known for all symmetry groups ( [51], [52] or [53] for a more rigorous derivation), and reads for the rank-n case…”

Section: B Free Additive Noisementioning

confidence: 99%

Rotational Invariant Estimator for General Noisy Matrices

Bun

Allez

Bouchaud

et al. 2016

IEEE Trans. Inform. Theory

108

View full text Add to dashboard Cite

We investigate the problem of estimating a given real symmetric signal matrix C from a noisy observation matrix M in the limit of large dimension. We consider the case where the noisy measurement M comes either from an arbitrary additive or multiplicative rotational invariant perturbation. We establish, using the Replica method, the asymptotic global law estimate for three general classes of noisy matrices, significantly extending previously obtained results. We give exact results concerning the asymptotic deviations (called overlaps) of the perturbed eigenvectors away from the true ones, and we explain how to use these overlaps to "clean" the noisy eigenvalues of M. We provide some numerical checks for the different estimators proposed in this paper and we also make the connexion with some well known results of Bayesian statistics. arXiv:1502.06736v2 [cond-mat.stat-mech]

show abstract

Section: B Free Additive Noisementioning

confidence: 99%

Rotational Invariant Estimator for General Noisy Matrices

Bun

Allez

Bouchaud

et al. 2016

IEEE Trans. Inform. Theory

108

View full text Add to dashboard Cite

show abstract

“…where • • • ρ denotes the average with respect to ρ(λ), while Extr θ {• • •} represents extremization with respect to θ [11]. This formula is analogous to the one known for ensembles of random square (symmetric) matrices [12], [13], [14], which is closely related to the R-transformation developed in free probability theory [15], [9], [16]. Several integral formulae for large random matrices related to Eq.…”

Section: A An Integral Formula For Large Random Rectangular Matricesmentioning

confidence: 94%

An integral formula for large random rectangular matrices and its application to analysis of linear vector channels

Kabashima

2008

Proceedings of the 6th Intl Symposium on Modeling and Optimization

View full text Add to dashboard Cite

A statistical mechanical framework for analyzing random linear vector channels is presented in a large system limit. The framework is based on the assumptions that the left and right singular value bases of the rectangular channel matrix H are generated independently from uniform distributions over Haar measures and the eigenvalues of H T H asymptotically follow a certain specific distribution. These assumptions make it possible to characterize the communication performance of the channel utilizing an integral formula with respect to H , which is analogous to the one introduced by Marinari et. al. in J. Phys. A 27, 7647 (1994) for large random square (symmetric) matrices. A computationally feasible algorithm for approximately decoding received signals based on the integral formula is also provided.

show abstract

“…Recently, it has been shown in [11] that the same analyses can be obtained without the need of Gaussian distribution assumption but a sub-Gaussian tail condition of the distribution is required. Indeed, thanks to the central limit theorem, one can show that the generating function (3) also yields the same result regardless of whether the Gaussian distribution assumption is considered or not, see [23,Section 5].…”

Section: Example 2 the Hopfield Modelmentioning

confidence: 98%

A theory of solving TAP equations for Ising models with general invariant random matrices

Opper¹,

Çakmak²,

Winther³

2016

J. Phys. A: Math. Theor.

114

View full text Add to dashboard Cite

We consider the problem of solving TAP mean field equations by iteration for Ising models with coupling matrices that are drawn at random from general invariant ensembles. We develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields an effective dynamics of a single variable trajectory. Our main novel contribution is the expression for the implicit memory term of the dynamics for general invariant ensembles. By subtracting these terms, that depend on magnetizations at previous time steps, the implicit memory terms cancel making the iteration dependent on a Gaussian distributed field only. The TAP magnetizations are stable fixed points if an AT stability criterion is fulfilled. We illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble.

show abstract

Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and free probability theory

Cited by 12 publications

References 19 publications

Rotational Invariant Estimator for General Noisy Matrices

Rotational Invariant Estimator for General Noisy Matrices

An integral formula for large random rectangular matrices and its application to analysis of linear vector channels

A theory of solving TAP equations for Ising models with general invariant random matrices

Contact Info

Product

Resources

About