非负矩阵分解中维数约简问题的研究

高维非负数据集的特征表示是模式识别、机器学习和图像处理等研究领域的核心课题，非负矩阵分解（NMF）是特征表示和低秩逼近的有效技术。NMF中的维数约简因子r 是直接影响特征表示的重要指标：如r取得过大，就加大计算工作量和存储空间;r取得太小，就丢失有用的信息而影响NMF的逼近效果。但目前几乎所有的NMF类算法都需要事先给定r的值，如何自适应得到r的值就成为NMF中具有挑战性的重要课题。本项目创新性地从讨论和研究数据空间形成的特殊锥（cone）的特性入手，依据我们课题组多年的工作积累，在构造基底矩阵W的同时确定r的值,然后通过解非负约束或非负稀疏约束最小二乘模型，通过最优化方法求得编码矩阵H。提出了改进NMF方法（INMF）和半NMF方法（SNMF）。该算法框架无需事先给定r的值，将问题转化为一个低阶凸问题，避免了NP-hard的困难，计算工作量明显减少，初步模拟结果体现了新算法的有效性

The feature represents of high demension negative data set is the core task in the field of pattern recognition, machine learning and image processing.Non-negative matrix factorization ( NMF ) is an effective technique for feature representation and low rank approximation.Dimension reduction factor r in NMF directly affect the calculation results of decomposition, such as r is too large, form data redundancy to increase the computational effort and memory space; r is too small, then the loss of useful information and influence the effect of NMF approximation. But the NMF algorithm almost all need to be given the value of r in advance, is an important problem which is in NMF. This project from a new point of view, the discussion and Research on the characteristics for a special cone will be formed by the data space. According to our research group in this work accumulation, and determine the value of r in the basis matrix W process, then have nonnegative constraints or non negative sparse constrained least square model, by the optimization method obtained coefficient matrix H. We will present the improved NMF method ( INMF ) and NMF method ( SNMF ). The algorithm framework without the given r value in advance, the problem is converted into a low order convex problem, avoid NP-hard difficult.The workload is reduced obviously, and the preliminary simulation results show the effectiveness of the new algorithm and the dimensionality advantage.