Spectral Density of Sparse Sample Covariance Matrices

Abstract

Applying the replica method of statistical mechanics, we evaluate the eigenvalue density of the large random matrix (sample covariance matrix) of the form $J=A^{\mathrm{T}}A$, where $A$ is an $M\times N$ real sparse random matrix. The difference from a dense random matrix is the most significant in the tail region of the spectrum. We compare the results of several approximation schemes, focusing on the behavior in the tail region.

22 pages, 4 figures, minor corrections made

References in corpus (2)

• Sparse random matrices: the eigenvalue spectrum revisited
• On the Largest Singular Values of Random Matrices with Independent Cauchy Entries

Cited by in corpus (19)

• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• Universality and the circular law for sparse random matrices
• Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices
• Spectral density of random graphs with topological constraints
• First eigenvalue/eigenvector in sparse random symmetric matrices: influences of degree fluctuation
• Power-law deformation of Wishart-Laguerre ensembles of random matrices
• A unifying model for random matrix theory in arbitrary space dimensions
• Equivalence of replica and cavity methods for computing spectra of sparse random matrices
• Spectral Density of Complex Networks with a Finite Mean Degree
• Bose-Einstein-Like condensation of deformed random matrix: A replica approach
• Localization in random bipartite graphs: numerical and empirical study
• Eigenvector localization as a tool to study small communities in online social networks
• Typical kernel size and number of sparse random matrices over GF(q) - a statistical physics approach
• Persistent individual bias in a voter model with quenched disorder
• Stability and dynamical properties of material flow systems on random networks
• Spectral Density of Complex Networks with Two Species of Nodes
• Sparse Random Block Matrices
• Top eigenpair statistics of diluted Wishart matrices