We present a natural generalization of the recent low rank + sparse matrix decomposition and consider the decomposition of matrices into components of multiple scales. Such decomposition is well motivated in practice as data matrices often exhibit local correlations in multiple scales. Concretely, we propose a multiscale low rank modeling that represents a data matrix as a sum of block-wise low rank matrices with increasing scales of block sizes. We then consider the inverse problem of decomposing the data matrix into its multiscale low rank components and approach the problem via a convex formulation. Theoretically, we show that under various incoherence conditions, the convex program recovers the multiscale low rank components either exactly or approximately. Practically, we provide guidance on selecting the regularization parameters and incorporate cycle spinning to reduce blocking artifacts. Experimentally, we show that the multiscale low rank decomposition provides a more intuitive decomposition than conventional low rank methods and demonstrate its effectiveness in four applications, including illumination normalization for face images, motion separation for surveillance videos, multiscale modeling of the dynamic contrast enhanced magnetic resonance imaging, and collaborative filtering exploiting age information.

中文翻译：

---

超越低秩+稀疏：多尺度低秩矩阵分解

我们提出了最近低秩+稀疏矩阵分解的自然概括，并考虑将矩阵分解为多个尺度的分量。这种分解在实践中很有动机，因为数据矩阵通常在多个尺度上表现出局部相关性。具体来说，我们提出了一种多尺度低秩建模，将数据矩阵表示为块大小逐渐增加的逐块低秩矩阵的总和。然后，我们考虑将数据矩阵分解为其多尺度低秩分量的逆问题，并通过凸公式来解决该问题。从理论上讲，我们表明，在各种不相干条件下，凸程序可以精确地或近似地恢复多尺度低秩分量。几乎，我们提供了选择正则化参数的指导，并结合循环旋转来减少块伪影。实验表明，多尺度低秩分解提供了比传统低秩方法更直观的分解，并证明了其在四种应用中的有效性，包括面部图像的照明归一化、监控视频的运动分离、动态对比度增强磁共振的多尺度建模成像和协同过滤利用年龄信息。