Spectral methods are widely used to estimate eigenvectors of a low-rank signal matrix subject to noise. These methods use the leading eigenspace of an observed matrix to estimate this low-rank signal. Typically, the entrywise estimation error of these methods depends on the coherence of the low-rank signal matrix with respect to the standard basis. In this work, we present a novel method for eigenvector estimation that avoids this dependence on coherence. Assuming a rank-one signal matrix, under mild technical conditions, the entrywise estimation error of our method provably has no dependence on the coherence under Gaussian noise (i.e., in the spiked Wigner model), and achieves the optimal estimation rate up to logarithmic factors. Simulations demonstrate that our method performs well under non-Gaussian noise and that an extension of our method to the case of a rank-$r$ signal matrix has little to no dependence on the coherence. In addition, we derive new metric entropy bounds for rank-$r$ singular subspaces under $\ell_{2,\infty}$ distance, which may be of independent interest. We use these new bounds to improve the best known lower bound for rank-$r$ eigenspace estimation under $\ell_{2,\infty}$ distance.

翻译：谱方法被广泛用于估计受噪声干扰的低秩信号矩阵的特征向量。这些方法利用观测矩阵的主特征空间来估计该低秩信号。通常，这些方法的逐项估计误差取决于低秩信号矩阵相对于标准基的相干性。本研究提出了一种新颖的特征向量估计方法，避免了这种对相干性的依赖。假设信号矩阵为秩一，在温和的技术条件下，我们的方法在加性高斯噪声下（即尖峰Wigner模型中）的逐项估计误差可证明与相干性无关，并以对数因子为界达到最优估计速率。仿真实验表明，我们的方法在非高斯噪声下表现良好，且将其推广至秩$r$信号矩阵的情形时几乎不依赖相干性。此外，我们推导了秩$r$奇异子空间在$\ell_{2,\infty}$距离下的新度量熵界，这可能具有独立的学术价值。我们利用这些新界改进了$\ell_{2,\infty}$距离下秩$r$特征空间估计的最佳已知下界。