凸与半非负矩阵分解的近点梯度方法研究

发布时间：2018-10-13 17:24

【摘要】：非负矩阵分解是高维非负数据降维中的一类重要数学模型,最近已经得到了大量研究观测。半非负矩阵分解与凸非负矩阵分解是针对一般数据而被Ding等[9]引入,他们是非负矩阵分解的推广,使得数据更有解释性。Ding等[9]提出了乘法更新策略,但对于大维数据,这种算法一般收敛很慢。在这篇文章中,我们对于半与凸非负矩阵分解给出了近点梯度方法,利用非光滑分析与Kurdyka-?ojasiewicz(KL)函数性质,我们证明了算法的收敛性。近些年一种Nesterov-类外推技术经常被采用去加速一些针对凸优化的梯度方法,所以我们尝试用这一技术去加速这非凸的半非负矩阵分解与凸非负矩阵分解问题,得到了很好的效果,使得算法的收敛速度得到了大幅度提高。在人工合成和真实数据集上的实验显示,我们的算法较之前的乘法更新策略在收敛速度方面提高至少10倍,在稀疏性和正交性方面略有提高,人工合成数据集上的分类实验显示,我们的算法是更有效的。
[Abstract]:Non-negative matrix factorization is an important mathematical model for dimensionality reduction of high-dimensional non-negative data. Semi-nonnegative matrix factorization and convex nonnegative matrix factorization are introduced by Ding et al. [9] for general data. They generalize non-negative matrix factorization, which makes the data more interpretive. Ding et al. [9] put forward multiplication updating strategy, but for large dimensional data, This algorithm generally converges very slowly. In this paper, we give the near-point gradient method for the semi-convex nonnegative matrix decomposition. By using the nonsmooth analysis and the properties of Kurdyka-?ojasiewicz (KL) functions, we prove the convergence of the algorithm. In recent years, a Nesterov- class extrapolation technique is often used to accelerate some gradient methods for convex optimization, so we try to accelerate the problem of nonconvex semi-negative matrix decomposition and convex non-negative matrix decomposition by using this technique, and get good results. The convergence speed of the algorithm is greatly improved. Experiments on artificial and real datasets show that our algorithm is at least 10 times faster in convergence speed and slightly more sparse and orthogonal than the previous multiplication update strategy. The classification experiments on synthetic data sets show that our algorithm is more effective.
【学位授予单位】：东北师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.21

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