We consider the problem of estimating a low-rank matrix from a noisy observed matrix. Previous work has shown that the optimal method depends crucially on the choice of loss function. In this paper, we use a family of weighted loss functions, which arise naturally for problems such as submatrix denoising, denoising with heteroscedastic noise, and denoising with missing data. However, weighted loss functions are challenging to analyze because they are not orthogonally-invariant. We derive optimal spectral denoisers for these weighted loss functions. By combining different weights, we then use these optimal denoisers to construct a new denoiser that exploits heterogeneity in the signal matrix to boost estimation with unweighted loss.

翻译：我们考虑从噪音观察的矩阵中估算低位矩阵的问题。 先前的工作表明, 最佳方法主要取决于损失功能的选择。 在本文中, 我们使用一组加权损失功能, 这些问题自然产生, 比如子矩阵分解、 与杂交性噪音分解、 与缺漏数据分解, 但是, 加权损失功能很难分析, 因为它们不是 orthoon- involution 。 我们为这些加权损失函数获得最佳的光谱储量器 。 通过将不同重量组合, 我们然后使用这些最佳的隐居器来构建一个新的脱色器, 利用信号矩阵中的异质性来增加非加权损失的估计 。