Top eigenpair statistics of diluted Wishart matrices

TL;DR

This paper uses the replica method to analytically study the statistics of the largest eigenpair of diluted sparse covariance matrices, providing insights into eigenvalue behavior and eigenvector structure in high-dimensional settings.

Contribution

It introduces a novel analytical approach using the replica method and population dynamics to characterize the top eigenpair of diluted Wishart matrices, extending previous models.

Findings

Analytical expressions for the average largest eigenvalue.

Efficient numerical solution via population dynamics.

Excellent agreement with direct diagonalization results.

Abstract

Using the replica method, we compute the statistics of the top eigenpair of diluted covariance matrices of the form $\mathbf{J} = \mathbf{X}^T \mathbf{X}$, where $\mathbf{X}$ is a $N\times M$ sparse data matrix, in the limit of large $N,M$ with fixed ratio and a bounded number of nonzero entries. We allow for random non-zero weights, provided they lead to an isolated largest eigenvalue. By formulating the problem as the optimisation of a quadratic Hamiltonian constrained to the $N$-sphere at low temperatures, we derive a set of recursive distributional equations for auxiliary probability density functions, which can be efficiently solved using a population dynamics algorithm. The average largest eigenvalue is identified with a Lagrange parameter that governs the convergence of the algorithm, and the resulting stable populations are then used to evaluate the density of the top eigenvector's components. We find excellent agreement between our analytical results and numerical results obtained from direct diagonalisation.