We study the problem of estimating a large, low-rank matrix corrupted by additive noise of unknown covariance, assuming one has access to additional side information in the form of noise-only measurements. We study the Whiten-Shrink-reColor (WSC) workflow, where a "noise covariance whitening" transformation is applied to the observations, followed by appropriate singular value shrinkage and a "noise covariance re-coloring" transformation. We show that under the mean square error loss, a unique, asymptotically optimal shrinkage nonlinearity exists for the WSC denoising workflow, and calculate it in closed form. To this end, we calculate the asymptotic eigenvector rotation of the random spiked F-matrix ensemble, a result which may be of independent interest. With sufficiently many pure-noise measurements, our optimally-tuned WSC denoising workflow outperforms, in mean square error, matrix denoising algorithms based on optimal singular value shrinkage which do not make similar use of noise-only side information; numerical experiments show that our procedure's relative performance is particularly strong in challenging statistical settings with high dimensionality and large degree of heteroscedasticity.

翻译：我们研究了在未知协方差加性噪声干扰下，估计大型低秩矩阵的问题，假设可以通过仅含噪声的测量获得额外的辅助信息。我们分析了"白化-收缩-重着色"（WSC）工作流，该流程先对观测数据施加"噪声协方差白化"变换，随后进行适当的奇异值收缩，最后执行"噪声协方差重着色"变换。研究表明，在均方误差损失下，WSC去噪工作流存在唯一且渐近最优的收缩非线性函数，并给出了其闭合形式解。为此，我们计算了随机尖峰F矩阵集合的渐近特征向量旋转，这一结果可能具有独立学术价值。在拥有足够多纯噪声测量的条件下，经最优调参的WSC去噪工作流在均方误差指标上优于基于最优奇异值收缩但未利用纯噪声辅助信息的矩阵去噪算法；数值实验表明，在高维度和强异方差性等具有挑战性的统计场景中，本文方法的相对性能尤为突出。