Eigenvalue and Eigenvector Approximation for Random Matrices Using Low-Degree Polynomials

We initiate the study of approximating the top eigenvalue and eigenvector of a random symmetric matrix $ A \in \mathbb{R}^{n\times n} $ using $ q(A)b $ where $q$ is a degree-$d$ polynomial and $b$ is a standard Gaussian vector independent of $A$. For spiked GOE $ Y = λvv^\top + X $, we identify $ d_