To exploit the information from two-dimensional structured data, two-dimensional principal component analysis (2-DPCA) has been widely used for dimensionality reduction and feature extraction. However, 2-DPCA is sensitive to outliers which are common in real applications. Therefore, many robust 2-DPCA methods have been proposed to improve the robustness of 2-DPCA. But existing robust 2-DPCAs have several weaknesses. First, these methods cannot be robust enough to outliers. Second, to center a sample set mixed with outliers using the L2-norm distance is usually biased. Third, most methods do not preserve the nice property of 2-DPCA (rotational invariance), which is important for learning algorithm. To alleviate these issues, we present a generalized robust 2-DPCA, which is named as 2-DPCA with \(\ell _{2,p}\)-norm minimization (\(\ell _{2,p}\)-2-DPCA), for image representation and recognition. In \(\ell _{2,p}\)-2-DPCA, \(\ell _{2,p}\)-norm is employed as the distance metric to measure the reconstruction error, which can alleviate the effect of outliers. Therefore, the proposed method is robust to outliers and preserves the desirable property of 2-DPCA which is invariant to rotational and well characterizes the geometric structure of samples. Moreover, most existing robust PCA methods estimate sample mean from database with outliers by averaging, which is usually biased. Sample mean are treated as an unknown variable to remedy the bias of computing sample mean in \(\ell _{2,p}\)-2-DPCA. To solve \(\ell _{2,p}\)-2-DPCA, we propose an iterative algorithm, which has a closed-form solution in each iteration. Experimental results on several benchmark databases demonstrate the effectiveness and advantages of our method.

为了利用二维结构化数据中的信息，二维主成分分析（2-DPCA）已被广泛用于降维和特征提取。但是，2-DPCA对实际应用中常见的异常值很敏感。因此，已经提出了许多鲁棒的2-DPCA方法来提高2-DPCA的鲁棒性。但是，现有的强大的2-DPCA具有多个缺点。首先，这些方法对异常值的鲁棒性不足。其次，使用L2范数距离将与离群值混合的样本集居中通常会产生偏差。第三，大多数方法没有保留2-DPCA（旋转不变性）的优良特性，这对于学习算法很重要。为了缓解这些问题，我们提出了一种通用的鲁棒2-DPCA，它被命名为2-DPCA，其\（\ ell _ {2，p} \）-norm最小化（\（\ ell _ {2，p} \）- 2-DPCA），用于图像表示和识别。在\（\ ell _ {2，p} \）- 2-DPCA中，使用\（\ ell _ {2，p} \）- norm作为距离度量来测量重构误差，这可以减轻离群值。因此，所提出的方法对于异常值是鲁棒的，并且保留了2-DPCA的理想特性，该特性对于旋转不变并且很好地表征了样品的几何结构。而且，大多数现有的鲁棒PCA方法通过平均来估计具有异常值的数据库中的样本均值，这通常是有偏差的。样本均值被视为未知变量，以纠正\（\ ell _ {2，p} \）- 2-DPCA中计算样本均值的偏差。解决\（\ ell _ {2，p} \）-2-DPCA，我们提出了一种迭代算法，该算法在每次迭代中都有一个封闭形式的解决方案。在几个基准数据库上的实验结果证明了我们方法的有效性和优势。