超越低秩加稀疏：多尺度低秩矩阵分解
Beyond Low Rank + Sparse: Multi-scale Low Rank Matrix Decomposition
AI总结：
本文提出了一种多尺度低秩矩阵分解方法，通过凸优化解决数据矩阵的多尺度分解问题，并在多个应用中验证了其有效性。
摘要：
我们提出了近期低秩加稀疏矩阵分解的自然推广，并考虑将矩阵分解为多个尺度的组件。这种分解在实践中得到了很好的动机，因为数据矩阵往往在多个尺度上表现出局部相关性。具体而言，我们提出了一种多尺度低秩建模，将数据矩阵表示为块状低秩矩阵之和，块的大小随着尺度增加。然后，我们考虑将数据矩阵分解为其多尺度低秩组件的逆问题，并通过凸优化方法解决。理论上，我们证明在各种不相干条件下，凸程序能够精确或近似地恢复多尺度低秩组件。实践中，我们提供了选择正则化参数的指导，并结合循环旋转以减少块效应。实验上，我们表明多尺度低秩分解比传统低秩方法更具直观性，并在四个应用中展示了其有效性，包括面部图像的照明归一化、监控视频的运动分离、动态对比增强磁共振成像的多尺度建模以及利用年龄信息的协同过滤。
英文摘要：
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 multi-scale 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 multi-scale low rank components and approach the problem via a convex formulation. Theoretically, we show that under various incoherence conditions, the convex program recovers the multi-scale low rank components \revised{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 multi-scale 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, multi-scale modeling of the dynamic contrast enhanced magnetic resonance imaging and collaborative filtering exploiting age information.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
