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：我们研究在附加噪声协方差未知的情况下，通过额外获取仅含噪声的测量作为辅助信息，对受加性噪声污染的大规模低秩矩阵进行估计的问题。我们分析了“白化-收缩-再着色”（WSC）工作流程：首先对观测数据施加“噪声协方差白化”变换，随后执行适当的奇异值收缩，最后进行“噪声协方差再着色”变换。我们证明，在均方误差损失函数下，WSC去噪工作流程存在唯一的渐近最优收缩非线性函数，并给出了其闭式解。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研究价值。当纯噪声测量样本量充足时，本文最优调参的WSC去噪工作流程在均方误差上优于基于最优奇异值收缩但不利用纯噪声辅助信息的矩阵去噪算法；数值实验表明，在高维度和高度异方差性的挑战性统计场景中，本方法的相对性能尤为突出。