基于RPCA模型的P范数优化算法

The Study of P-norm Optimization Algorithm Based on RPCA Model

在图像修复和视频处理中,低秩矩阵恢复有着非常广泛的应用.RPCA模型是低秩矩阵恢复的经典模型,其基本思想是将一个数值矩阵分解为一个低秩矩阵与一个稀疏矩阵和的形式再进行求解.然而,RPCA问题是NP难的,一个通用的处理方式就是将RPCA模型中矩阵的秩函数和0L范数分别松弛为矩阵的核范数和L_1范数,从而将其近似转化为凸优化问题来求解,但这种由凸优化近似方法得到的解在相对较弱的非相干性条件下会使原始问题的解退化.针对这个问题,本文首先提出一种更接近于原始问题的非凸近似模型,即用矩阵的Schatten-p范数和L_p范数(0<p<1)分别代替矩阵的秩函数和L_0范数,然后针对提出的非凸近似模型,进一步给出有效的优化算法,最后,在人工数据集和真实图像数据集上进行实验,结果表明,所提出的模型是有效的.

The model of low-rank matrix recovery has been widely used in image in painting and video processing field. RPCA model is a classical model of low rank matrix recovery. Its basic idea is to decompose a numerical matrix into the form of a low-rank matrix and a sparse matrix sum before it is solved. However, RPCA problem is NP-hard. A commonly-used processing mode is to loose the matrix rank function and the L_0 norm from the RPCA model respectively into the matrix nuclear norm and the L_1 norm. Thus, the problem is solved through the way to transform its approximation into the convex optimization problem. However, the solution of the convex optimization approximation approach degrades the solution of original problem under the relatively weak incoherent condition. This paper firstly proposes a non-convex approximation model toward the problem, namely, applying the matrix Schatten-p norm andpL norm( 0 <p <1) to replace the matrix rank function and L_0 norm. And then the further effective optimization algorithm is given based on the non-convex approximation model. Lastly, the experiment based on the synthetic datasets and the real image datasets is made. It turns out that the model proposed is valid and effective.