AI 中文总结
本研究确定了随机矩阵顶部特征值与特征向量逼近的临界多项式次数阈值，建立了谱方法与低次多项式算法的精确联系，并给出了精确的渐近逼近精度。
AI 中文摘要
我们开创了对随机对称矩阵 $ A \in \mathbb{R}^{n\times n} $ 的顶部特征值和特征向量使用 $ q(A)b $ 进行逼近的研究，其中 $q$ 是次数为 $d$ 的多项式，$b$ 是独立于 $A$ 的标准高斯向量。对于尖峰GOE $ Y = \lambda vv^\top + X $，我们确定 $ d_\star = \frac{\log(n)}{2\log(\lambda)} $ 是临界次数阈值，超过该阈值即可对顶部特征值和特征向量进行精确逼近。这锐化了谱方法可通过 $ O(\log(n)) $ 步幂迭代实现的普遍认知，并为谱方法与低次多项式算法（所有多项式时间算法的流行代理）之间提供了精确联系。对于GOE $X$，我们确定 $ d_\star = n^{1/3+o(1)} $ 是顶部特征向量逼近的临界次数阈值，而常数次数足以用于顶部特征值逼近。此外，在 $ d/n^{1/3} $ 收敛于正有限常数的极限中，我们以期望平方重叠的形式计算了精确的渐近特征向量逼近精度。这些结果显著改进了随机化数值线性代数中对确定性数据矩阵的预测，即幂方法的迭代次数由逆谱间隙控制。在技术上，我们的分析利用了切比雪夫多项式的极值性质，并借鉴了随机矩阵理论的丰富文献。
英文摘要
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 with random initialization 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.
