Marginals of a spherical spin glass model with correlated disorder

Barbier, Jean; Sáenz, Manuel

doi:10.48550/arxiv.2112.02066

Cited by 2 publications

(2 citation statements)

References 23 publications

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“…Because of the simple structure of the Hamiltonian in the spherical case, the computation of the free energy is closely tied to the behavior of the eigenvalues of a GOE matrix, which has been the studied extensively in random matrices. Random matrix techniques have been applied to study the fluctuations of the free energy and corresponding phase transtions in [6,7,5], the connection the large deviations of the top eigenvalue in [54], and the marginals of spherical spin glasses with correlated disorder matrices in [11].…”

Section: Introductionmentioning

confidence: 99%

Spherical Integrals of Sublinear Rank

Husson¹,

Ko²

2022

Preprint

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We consider the asymptotics of k-dimensional spherical integrals when k = o(N ). We prove that the o(N )-dimensional spherical integrals are approximately the products of 1-dimensional spherical integrals. Our formulas extend the results for k-dimensional spherical integrals proved by Guionnet and Maïda in [30] and Husson and Guionnet in [35] which are only valid for k finite and independent of N . These approximations will be used to prove a large deviation principle for the joint 2k(N ) extreme eigenvalues for sharp sub-Gaussian Wigner matrices and for additive deformations of GOE/GUE matrices. Furthermore, our results will be used to compute the free energies of spherical SK vector spin glasses and the mutual information for matrix estimation problems when the dimensions of the spins or signals have sublinear growth.

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Section: Introductionmentioning

confidence: 99%

Spherical Integrals of Sublinear Rank

Husson¹,

Ko²

2022

Preprint

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“…• Our analysis can be mainstreamed once we have identified a key integral that we refer to as the inhomogeneous spherical integral. This exactly solvable integral is a generalization of the standard low-rank spherical integral appearing in random matrix theory (as it is related to the R-transform) [93], in spin-glasses [89,86,84,42,20], the theory of large-deviations for matrix-valued stochastic processes [54,55] and matrix models in high-energy physics [56,62,52]. Given the breadth of applications of this integral, we foresee that the generalization we propose and analyze in Section 2 may have applications well beyond the present setting, for the study of models where rotationally invariant matrices with non-independent matrices appear.…”

Section: Information-theoretic Resultsmentioning

confidence: 99%

Bayes-optimal limits in structured PCA, and how to reach them

Barbier¹,

Camilli²,

Mondelli³

et al. 2022

Preprint

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We study the paradigmatic spiked matrix model of principal components analysis, where the rank-one signal is corrupted by additive noise. While the noise is typically taken from a Wigner matrix with independent entries, here the potential acting on the eigenvalues has a quadratic plus a quartic component. The quartic term induces strong correlations between the matrix elements, which makes the setting relevant for applications but analytically challenging. Our work provides the first characterization of the Bayes-optimal limits for inference in this model with structured noise. If the signal prior is rotational-invariant, then we show that a spectral estimator is optimal. In contrast, for more general priors, the existing approximate message passing algorithm (AMP) falls short of achieving the information-theoretic limits, and we provide a justification for this sub-optimality. Finally, by generalizing the theory of Thouless-Anderson-Palmer equations, we cure the issue by proposing a novel AMP which matches the theoretical limits. Our information-theoretic analysis is based on the replica method, a powerful heuristic from statistical mechanics; instead, the novel AMP comes with a rigorous state evolution analysis tracking its performance in the high-dimensional limit. Even if we focus on a specific noise distribution, our methodology can be generalized to a wide class of trace ensembles, at the cost of more involved expressions.

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Marginals of a spherical spin glass model with correlated disorder

Cited by 2 publications

References 23 publications

Spherical Integrals of Sublinear Rank

Spherical Integrals of Sublinear Rank

Bayes-optimal limits in structured PCA, and how to reach them

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