12:00-12:30 lunch；12:30-13:30 Talk

Abstract: In this talk, we study a curious phenomenon reflecting the interplay between asymmetry and the

spectral methods. Suppose we are interested in a rank-1 and symmetric matrix n-by-n matrix M*, yet only

a randomly perturbed version M is observed. The noise matrix M−M* is composed of independent and

zero-mean entries and is not symmetric. This might arise, for example, when we have two independent

samples for each entry of M* and arrange them into an asymmetric data matrix M. The aim is to estimate

the leading eigenvalue and eigenvector of M*. Somewhat unexpectedly, our findings reveal that the leading

eigenvalue of the data matrix M can be √n times more accurate than its leading singular value in eigenvalue

estimation. Further, the perturbation of any linear form of the leading eigenvector of M (e.g. entrywise

eigenvector perturbation) is provably well-controlled. We further provide partial theory for the more

general rank-r case; this allows us to accommodate the case when M* is rank-1 but asymmetric, by

considering eigen-decomposition of the associated rank-2 dilation matrix. The takeaway message is this:

arranging the data samples in an asymmetric manner and performing eigen-decomposition (as opposed to SVD) could sometimes be quite beneficial.

欢迎各位同学积极参加，报名链接https://www.wenjuan.com/s/jYnyu2A/， 报名截止时间为3月15日上午10:00，我们将为报名同学提供午餐。