面向鲁棒特征提取的矩阵秩极小化问题研究

Robust feature extracting from the incomplete and corrupted high-dimensional data is one of hot topic in pattern recognition by employing the sparsity. Based on the assumption of low rank, it can be viewed as a rank minimization problem. The rank minimization problem is mainly achieved by the nuclear norm minimization problem, which characterizing the noise based on the assumption of independent identically distributed. However there exists the structural noise in application and its elements are relative and not independent. This project is focus on the research for generalized nuclear norm for characterizing structural noise and its application for feature extracting. (1) This project tries to propose matrix variate distribution of structural noise and the relationship between matrix variate distribution and generalized nuclear norm. (2) This project tries to propose a generalized nuclear norm based structural noise characterizing method. The generalized nuclear norm can approximate the rank function more than nuclear norm. (3) This project tries to propose the subspace learning based low rank decomposition. The method distinguishes the dominating component and secondary components, so as to realize the robust feature extraction. .Compared with traditional feature extraction methods, the research can improve the robustness to structural noise. Moreover, the research provides theory support for the related research areas.

利用高维数据中存在的稀疏性,如何从不完整的、带有损毁的数据中鲁棒地提取特征是模式识别领域的热点问题之一。该问题的解决方法往往基于低秩假设,从而将问题建模为秩极小化问题。对秩极小化问题的研究主要通过研究核范数极小化问题实现。目前的核范数极小化问题在刻画噪声时大多是基于独立同分布假设的。实际问题中经常面临结构噪声,这类噪声的元素之间具有相关性,不再符合独立同分布假设。本项目拟研究广义核范数在刻画结构噪声时的机理及其在特征抽取中的有效性。(1) 给出结构噪声的矩阵变量概率分布模型，及广义核范数与多元概率分布之间的关系。(2)提出基于广义核范数的结构噪声刻画方法。广义核范数比核范数能更有效地近似秩函数。(3)提出基于低秩分解的子空间学习。通过区别对待数据的主要分量和次要分量实现特征的鲁棒提取。.预期成果与传统的特征提取方法相比，在结构噪声的鲁棒性方面会有较大改善，进而为相关领域的研究提供理论支撑。

利用高维数据中存在的稀疏性,如何从不完整的、带有损毁的数据中鲁棒地提取特征是模式识别领域的热点问题之一。该问题的解决方法往往基于低秩假设,从而将问题建模为秩极小化问题。对秩极小化问题的研究主要通过研究核范数极小化问题实现。以往的核范数极小化问题在刻画噪声时大多是基于独立同分布假设的。实际问题中经常面临结构噪声,这类噪声的元素之间具有相关性,不再符合独立同分布假设。本项目着重研究了广义核范数在刻画结构噪声时的机理及其在特征抽取中的有效性问题：第一，给出了结构噪声的矩阵变量概率分布模型和广义核范数与多元概率分布之间的关系；第二，提出了基于广义核范数的结构噪声刻画方法；第三，提出了基于低秩分解的子空间学习。通过区别对待数据的主要分量和次要分量实现特征的鲁棒提取。研究成果与传统的特征提取方法相比，在结构噪声的鲁棒性方面有较大改善，为相关领域的研究提供了理论支撑。