基于均衡张量与非凸函数的高精度低秩张量恢复方法

A High Precision Low Rank Tensor Recovery Method Based on Balanced Ten⁃sor and Non-convex Function

基于最小化核范数之和的低秩张量恢复方法已在许多领域取得广泛应用,但在某些应用场景,如彩色视频恢复中,其恢复结果不够理想,尤其是当已知元素数目较少时,恢复结果更差。导致该问题出现的一个根本原因是张量的不均衡结构,即张量中某些模的维度远小于其他模,这使得其展开矩阵是一个极度不均衡且满秩或高秩的矩阵。此外,秩的凸松弛则是导致该问题出现的另一个重要原因,最新研究表明,一些非凸函数可以比核范数更好地逼近秩。基于上述考虑,首先,引入一个重构操作将不均衡张量重构为更加均衡的张量,该重构操作能在保留张量低秩特性的同时减少算法准确恢复所需的已知元素数目;其次,将非凸logDet函数引入低秩张量恢复算法中,以取得比核范数更好的秩逼近效果;最后,通过与其他经典算法在彩色视频上开展对比实验,充分验证了本方法的有效性。

The classical low-rank tensor recovery method based on the minimization of the sum of the nuclear norms has been widely used in many fields.While,the results are not satisfactory in some specific scenarios,such as color vid⁃eo recovery,especially when the number of observations is small.One of the root causes lies in the unbalanced struc⁃ture of the tensor,in which the dim sizes of some modes are much smaller than others,leading to extremely unbalanced and actually full or high rank unfolding matrices.Another reason should be attributed to the convex substitution of rank,and recent researches have demonstrated that some non-convex functions could approximate rank better than nuclear norm.To address the problem,this paper reconstructs a more balanced tensor through a reshaping operator,which retains the low rank property.Furthermore,a non-convex function is brought into tensor recovery as a better ap⁃proximation of rank.Finally,experiments are carried out on color videos and the experimental results validate the ef⁃fective of the method.