🤖 AI Summary
This study investigates the low-degree polynomial approximation of the leading eigenpair of random symmetric matrices. Focusing on the Spiked Gaussian Orthogonal Ensemble (GOE) and standard GOE models, it integrates exact spectral methods with the low-degree algorithmic framework. By leveraging the extremal properties of Chebyshev polynomials and random matrix theory, this work rectifies prevailing misconceptions regarding the required number of iterations in classical power methods. It establishes a critical degree threshold for approximating the leading eigenpair and derives an exact expression for the asymptotic overlap. The resulting theoretical predictions significantly improve upon existing bounds, offering new insights into the fundamental limits of polynomial-based algorithms for random matrix computations.
📝 Abstract
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_\star = \frac{\log(n)}{2\log(λ)} $ to be the critical degree threshold above which accurate approximation of the top eigenvalue and eigenvector is possible. This sharpens the common belief that spectral methods can be implemented by $ O(\log(n)) $-step power iterations and offers a precise connection between spectral methods and low-degree polynomial algorithms, a popular proxy for all polynomial-time algorithms. For GOE $X$, we identify $ d_\star = n^{1/3+o(1)} $ to be the critical degree threshold for top eigenvector approximation, whereas constant degree suffices for top eigenvalue approximation. Moreover, in the limit where $ d/n^{1/3} $ converges to a positive finite constant, we compute the exact asymptotic eigenvector approximation accuracy in terms of the expected squared overlap. These results significantly improve upon predictions made in randomized numerical linear algebra for deterministic data matrices that the iteration count of power methods is governed by the inverse spectral gap. Technically, our analyses leverage extremal properties of Chebyshev polynomials and draw upon the rich literature of random matrix theory.