ASYMMETRY HELPS: EIGENVALUE AND EIGENVECTOR ANALYSES OF ASYMMETRICALLY PERTURBED LOW-RANK MATRICES.

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $M^{\star }\in {ℝ}^{n\times n}$ , yet only a randomly perturbed version M is observed. The noise matrix M - M ^⋆ is composed of independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise if, for example, we have two independent samples for each entry of M ^⋆ and arrange them in an asymmetric fashion. The aim is to estimate the leading eigenvalue and the leading eigenvector of M ^⋆. We demonstrate that the leading eigenvalue of the data matrix M can be $O\left(\sqrt{n}\right)$ times more accurate (up to some log factor) than its (unadjusted) leading singular value of M in eigenvalue estimation. Moreover, the eigen-decomposition approach is fully adaptive to heteroscedasticity of noise, without the need of any prior knowledge about the noise distributions. In a nutshell, this curious phenomenon arises since the statistical asymmetry automatically mitigates the bias of the eigenvalue approach, thus eliminating the need of careful bias correction. Additionally, we develop appealing non-asymptotic eigenvector perturbation bounds; in particular, we are able to bound the perterbation of any linear function of the leading eigenvector of M (e.g. entrywise eigenvector perturbation). We also provide partial theory for the more general rank-r case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be quite beneficial.