论文标题
投影强大的PCA具有应用程序以平滑图像恢复
Projected Robust PCA with Application to Smooth Image Recovery
论文作者
论文摘要
在目标矩阵具有某些内在结构的假设下，研究了大多数高维矩阵恢复问题。对于与图像数据相关的矩阵恢复问题，近似低级别和平滑度是两个最常见的结构。对于大约较低的矩阵恢复，可靠的主成分分析（PCA）经过充分研究并被证明是有效的。对于光滑的矩阵问题，2D Fused Lasso和其他基于总变化的方法发挥了基本作用。尽管低级别和平滑度都是图像数据分析的关键假设，但是两条研究线的相互作用非常有限。通过利用这两个功能，我们在本文中开发了一个名为“强大PCA（PRPCA）”的框架，在该框架下，低级矩阵被投影到光滑矩阵的空间上。因此，可以将大量的图像矩阵分解为低级别和光滑的组件以及稀疏组件。这种分解的一个关键优势是核心低级数的尺寸可以大大降低。因此，我们的框架能够解决许多低级矩阵问题的有问题的瓶颈：大型矩阵上的奇异价值分解（SVD）。从理论上讲，我们提供PRPCA的明确统计恢复保证，并将鲁棒的PCA作为特殊情况。
Most high-dimensional matrix recovery problems are studied under the assumption that the target matrix has certain intrinsic structures. For image data related matrix recovery problems, approximate low-rankness and smoothness are the two most commonly imposed structures. For approximately low-rank matrix recovery, the robust principal component analysis (PCA) is well-studied and proved to be effective. For smooth matrix problem, 2d fused Lasso and other total variation based approaches have played a fundamental role. Although both low-rankness and smoothness are key assumptions for image data analysis, the two lines of research, however, have very limited interaction. Motivated by taking advantage of both features, we in this paper develop a framework named projected robust PCA (PRPCA), under which the low-rank matrices are projected onto a space of smooth matrices. Consequently, a large class of image matrices can be decomposed as a low-rank and smooth component plus a sparse component. A key advantage of this decomposition is that the dimension of the core low-rank component can be significantly reduced. Consequently, our framework is able to address a problematic bottleneck of many low-rank matrix problems: singular value decomposition (SVD) on large matrices. Theoretically, we provide explicit statistical recovery guarantees of PRPCA and include classical robust PCA as a special case.
