Spherical Integrals of Sublinear Rank

Abstract

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 [29] and Husson and Guionnet in [34] 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.

arXiv commentCorrected the argument in Section 3.1. Removed an assumption in Proposition 2.8. Corrected typos

References in corpus (18)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• An Extended Variational Principle for the SK Spin-Glass Model
• Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula
• MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel
• Mutual Information in Rank-One Matrix Estimation
• On the overlap in the multiple spherical SK models
• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with ferromagnetic interaction
• Fundamental limits of detection in the spiked Wigner model
• Ferromagnetic to paramagnetic transition in spherical spin glass
• Perturbative construction of mean-field equations in extensive-rank matrix factorization and denoising
• Statistical limits of dictionary learning: random matrix theory and the spectral replica method
• Character expansion method for the first order asymptotics of a matrix integral
• An inference problem in a mismatched setting: a spin-glass model with Mattis interaction
• On Horn's Problem and its Volume Function
• The Crisanti-Sommers Formula for Spherical Spin Glasses with Vector Spins
• Free energy landscapes in spherical spin glasses
• Optimal denoising of rotationally invariant rectangular matrices
• Statistical inference of finite-rank tensors