A Tensor Recovery Method Based on Exponentialized Kernel Norm and Mixed Singular Value Truncation

Abstract

The invention discloses a tensor recovery method based on exponential nuclear norm and mixed singular value truncation, which mainly includes the following steps: firstly, a new definition of tensor rank is proposed, and the maximum rank of the expanded matrix under different tensor modes is Value, the logarithm of the nuclear norm exponent sum is used to approximate the tensor rank definition, and it is transformed into a convex function; secondly, in order to eliminate the correlation of the matrix expanded in different modes of the tensor, a series of auxiliary variables are introduced to replace the expanded matrix, and transform the constraints into augmented Lagrangian functions using the Lagrangian multiplier method; finally, the alternating direction method is used to iteratively optimize various variables in the augmented Lagrangian function until convergence. Among them, the present invention is a general method for optimizing variables in the exponent sum of the nuclear norm. Compared with other classical tensor recovery methods, this method can better describe the internal structure of high-dimensional data, thereby obtaining more Good recovery results.

Description

technical field [0001] 本发明涉及计算机模式识别技术领域,具体涉及一种基于指数化核范数与混合奇异值截断的张量恢复方法。 Background technique [0002] 张量恢复(tensor completion),即高维矩阵的恢复问题,对于一个部分元素缺失的高维矩阵,通过观察其已有位置的元素,从而恢复出缺失部分元素的一般性问题,是计算机模式识别领域中的研究热点之一,被广泛应用于图像去噪、图像恢复、信息推荐系统等众多领域。总的来说,大多数现有的张量恢复方法是基于低秩结构假设,即要求恢复的缺失元素使得整个张量的秩尽可能的小。目前有两种传统的定义张量秩的方法:基于张量的CP(CANDECOMP / PARAFAC)分解方法(CP秩)和Tucker分解方法(Tucker秩)。具体来说,CP秩可以定义为:用秩一张量(rank-one tensor)之和来表示给定张量需要的秩一张量的最小个数。Tucker秩可以定义为:不同模态下展开矩阵的秩的线性加权。无论是CP秩还是Tucker秩,最小化该张量秩的优化问题被证明是一个NP难问题。 [0003] 为了解决上述问题,Gandy等人采用不同模态下展开矩阵秩的和来表示张量的秩,在计算过程中,将展开矩阵的秩用矩阵的核范数来近似代替。Signoretto等人提出一种Shatten-p范数来代替展开矩阵的核范数,并由此定义了张量的秩,最后讨论了该方法与核范数之间的关系。随后,Liu等人采用不同模态下展开矩阵核范数的线性加权来近似代替Tucker秩,并将该方法应用于图像恢复和医学图像去噪。最后,Tomioka等人对张量恢复方法进行了总结,认为有两种方式可以实现张量恢复:(1)将张量按某一个模态展开成二阶矩阵,可以将张量恢复问题转化为了二阶矩阵的恢复问题;(2)采用不同模态下展开矩阵核范数的线性加权来近似代替Tucker秩。 [0004] 可以看出,上述方法的目的都是寻找张量Tucker秩的近似。然而Tucker秩定义的几何意义不清晰,并且张量不同模态下的权重难以选择,如果某个模态下展开矩阵的秩很大,而其对应的权重很小,那么上述定义将无法正确地描述张量的秩结构,从而导致张量低秩恢复效果不够理想。张量的CP秩是对矩阵秩的一个推广,它的几何意义比Tucker秩更为明确,然而直接优化CP秩...

AI Technical Summary

Problems solved by technology

However, the geometric meaning of the Tucker rank definition is not clear, and it is difficult to choose the weights of the tensor in different modes. If the rank of the expanded matrix in a certain mode is large, but the corresponding weight is small, then the above definition will not be correct. Describe the rank structure of the tensor, which leads to the unsatisfactory low-rank recovery effect of the tensor

The CP rank of the tensor is an extension of the matrix rank, and its geometric meaning is clearer than the Tucker rank. However, directly optimizing the CP rank is a very difficult problem, and it is even difficult to obtain a local optimum.

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