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