Overlaps between eigenvectors of spiked, correlated random matrices: From matrix principal component analysis to random Gaussian landscapes

Abstract

We consider pairs of Gaussian orthogonal ensemble matrices which are correlated with each others, and subject to additive and multiplicative rank-one perturbations. We focus on the regime of parameters in which the finite-rank perturbations generate outliers in the spectrum of the matrices. We investigate the statistical correlation (i.e., the typical overlap) between the eigenvectors associated to the outlier eigenvalues of each matrix in the pair, as well as the typical overlap between the outlier eigenvector of one matrix with the eigenvectors in the bulk of the spectrum of the other matrix. We discuss implications of these results for the signal recovery problem for spiked matrices, as well as for problems of high-dimensional random landscapes.

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“…The presence of the special row and column can give rise to subleading contributions to the eigenvalue density: these contributions correspond to eigenvalues that do not belong to the support of the semicircular law (and are said to be 'isolated'), and whose typical value depends on the parameters ∆, µ a governing the statistics of the entries of the special row and column. As argued in [33], the fact that ∆ ⩽ σ (as can be easily verified to be the case here), implies that only one isolated eigenvalue can exist for these matrices. Such eigenvalue exists whenever…”

“…It is shown in [33] that equation (E12) is indeed positive on the Right Hand Side, as it should be. Moreover, whenever the squared overlap is non-zero, the eigenvector is aligned with the direction of the finite-rank perturbation, meaning that e a iso • e a N−1 > 0.…”

“…Let us neglect the term proportional to the identity, which corresponds to a constant shift of the whole spectrum, and work with HA (σ a ). The spectral properties of these kind of matrices are discussed in detail in [33]. To leading order in the size of the matrix, the eigenvalue density ρ a N (λ) of both matrices is not affected by the presence of the special line and column, and it just coincides with the eigenvalue density of the GOE block, i.e.…”

“…Whenever the isolated eigenvalue exists, its eigenvector e a iso has a projection on the vector e N−1 (σ a ), corresponding to the special line and column of the matrix, which remains of O(1) when N is large. The typical value of this projection has been computed in [27,33] and reads:…”

“…As we argue in appendix G, to determine its typical behavior for large N one needs to compute the typical value of the overlap between arbitrary eigenvectors of the two correlated matrices. This random matrix problem is discussed in recent literature for standard random matrix ensembles [31,32]; for the particular type of random matrix ensembles describing the statistics of the Hessians H(σ a ), this overlap function has been computed explicitly in [33], and the results are recalled in appendix G. Below, we discuss the properties of the energy profiles that are obtained making use of these results.…”