Rank-One Signal Recovery in Sparse Wishart Noise

We study the high-dimensional recovery of a signal vector x in the presence of sparse Wishart-like noise. We define an N × N matrix A = J+(θ/N)xxtop, where xxtop is the rank-one deformation of the random noise matrix J. We consider a Wishart-like matrix J=Xtop X, where X is a sparse M × N random matrix with entries Xij = cijWij, with cij regulating the density of non-zero elements, and Wij the bond weights. Using the replica method, we compute analytically the top eigenpair statistics of A, and their dependence on the signal strength θ, the rectangularity ratio α=√M/N, and the average connectivity of the noise. The spectral observables are expressed in terms of a system of Recursive Distributional Equations, which are efficiently solved via a Population Dynamics algorithm. They allow us to compute the average largest eigenvalue langleλ1rangleA, the average top eigenvector component density, and the average overlap between the top eigenvector of A and x. We identify a critical threshold θcrit--depending on the average connectivity of the noise--that marks a BBP-like phase transition: below this value, langleλ1rangleA is unaffected by the signal, and the overlap vanishes. Thus, the signal is not recoverable from the top eigenvector of A. For θ>θcrit, the signal-related outlier eigenvalue becomes langleλ1rangleA and the overlap is nonzero, allowing for recovery of the signal. The results are in excellent agreement with numerical diagonalisation. We show that in the dense limit, the recovery threshold and eigen-statistics converge to the results predicted by the classical BBP transition for additive rank-one deformations of dense Wishart matrices.