一类可正交分解四阶共轭部分对称张量的性质和低秩逼近  

The properties and the low-rank approximation of a class of unitary decomposable fourth-order conjugate partial symmetric tensors

作　　者：唐云飞 杨庆之[1] Yunfei Tang;Qingzhi Yang

机构地区：[1]南开大学数学科学学院,核心数学与组合数学实验室,天津300071

出　　处：《中国科学:数学》2025年第2期301-322,共22页Scientia Sinica:Mathematica

基　　金：国家自然科学基金(批准号:12071234)资助项目。

摘　　要：本文将酉可分解张量的概念推广到广义酉可分解,研究了四阶广义酉可分解共轭部分对称(conjugate partial symmetric,CPS)张量的性质和低秩逼近,证明了四阶广义酉可分解CPS张量的CPS秩一和CPS秩二逼近问题等价于带一范数约束的复二次优化问题.对两类广义酉可分解张量,本文证明了可以通过序列秩一逼近和序列秩二逼近算法精确恢复原张量.对CP(CANDECOMP/PARAFAC)秩小于阶数的四阶部分对称(partial symmetric,PS)张量,本文证明了其PS秩等于CP秩,刻画了低秩四阶CPS张量的特征.In this paper,we extend the notion of unitary decomposable tensors to general unitary decomposable tensors,and study the properties and the low-rank approximation of fourth-order general unitary decomposable tensors with conjugate partial symmetric(CPS)property.We show that the CPS rank one approximation problem for fourth-order general unitary decomposable CPS tensors is equivalent to maximizing a complex quadratic form with 1-norm constraint.For two classes of fourth-order general unitary decomposable CPS tensors,we show that the successive rank one approximation algorithm along with the successive rank two approximation algorithm can exactly recover the original tensor.For fourth-order partial symmetric(PS)tensors,we prove that when the rank of the tensor is smaller than its order,then the CP rank and the PS rank coincide.Based on this,we characterize the shape of all rank 2 fourth-order CPS tensors.

关 键 词：共轭部分对称 正交分解 张量分解 低秩逼近 

分 类 号：O17[理学—数学]