解张量分解问题的信赖域交替最小二乘法

【作者】 张晓飞；

【导师】 韩德仁；

【作者基本信息】 南京师范大学 ， 计算数学， 2014， 硕士

【摘要】 张量分解问题在众多领域(如信号处理,图像分析,生物医疗等),有着广泛的应用,对其理论和算法的研究也引起了众多学者越来越多的重视.本文研究对求解该问题的最流行算法--交替最小二乘法进行适当的改进,证明其全局收敛性.为此,我们引入优化方法中的信赖域技术,提出基于信赖域的交替最小二乘方法求解张量分解问题.利用已有的信赖域半径调整手段,本文给出了参数的自适应选取准则.在非常一般的假设前提下,证明了算法的全局收敛性,解决了交替最小二乘法的收敛性问题.同时本文的分析也可用于正则化交替最小二乘法,证明正则化交替最小二乘法也有全局收敛性,而不仅仅是弱收敛性.为了提高算法的效率,本文也对算法进行了加速,即通过外推获得新的迭代点.为了验证理论分析结果,本文将算法应用到氨基酸荧光数据分解,并与基本的和正则的交替最小二乘法进行了比较.数值结果表明,新的方法不论在迭代步数还是迭代时间上都远远优于基本的交替最小二乘法.更多还原

【Abstract】 Tensor decomposition is an important tool in lots of fields, such as signal processing, graph analysis, biomedical engineering, etc, and many scholars put more and more attentions on its theory and algorithms. This paper aims at modifying the most pop-ular algorithm for tensor decomposition--alternating least squares (ALS), such that the global convergence of the resulting algorithm can be guaranteed under mild conditions. To this end, we combine the idea of trust region with alternating least squares, generating trust region based alternating least squares algorithm. Adopt-ing a simple technique for adjusting the trust region radius, this paper presents an adaptive parameter skill. Under very general assumptions, we prove the global convergence of the new algorithm. We also show that with the same analysis, the global convergence of the regularized alternating least squares algorithm can also be proved. In order to accelerate the new algorithm, we utilize the extrapolation to get the new iteration point. At last, we apply the new algorithm in the amino acid fluorescence data decomposition, and compare it with ALS and regularized ALS. The numerical results show that the new algorithm is more superior than the basic ALS and the regularized ALS, in the sence of both number of iterations and CPU time.更多还原