Feature selection plays a critical role in data mining, driven by increasing feature dimensionality in target problems. In this paper, we propose a new criterion for discriminative feature selection, worst-case discriminative feature selection (WDFS). Unlike Fisher Score and other methods based on the discriminative criteria considering the overall (or average) separation of data, WDFS adopts a new perspective called worst-case view which arguably is more suitable for classification applications. Specifically, WDFS directly maximizes the ratio of the minimum of between-class variance of all class pairs over the maximum of within-class variance, and thus it duly considers the separation of all classes. Otherwise, we take a greedy strategy by finding one feature at a time, but it is very easy to implement. Moreover, we also utilize the correlation between features to help reduce the redundancy and extend WDFS to uncorrelated WDFS (UWDFS). To evaluate the effectiveness of the proposed algorithm, we conduct classification experiments on many real data sets. In the experiment, we respectively use the original features and the score vectors of features over all class pairs to calculate the correlation coefficients, and analyze the experimental results in these two ways. Experimental results demonstrate the effectiveness of WDFS and UWDFS.

Most existing robust principal component analysis (PCA) involve mean estimation for extracting low-dimensional representation. However, they do not get the optimal mean for real data, which include outliers, under the different robust distances metric learning, such as L1-norm and L2,1-norm. This affects the robustness of algorithms. Motivated by the fact that the variance of data can be characterized by the variation between each pair of data, we propose a novel robust formulation for PCA. It avoids computing the mean of data in the criterion function. Our method employs L2,p-norm as the distance metric to measure the variation in the criterion function and aims to seek the projection matrix that maximizes the sum of variation between each pair of the projected data. Both theoretical analysis and experimental results demonstrate that our methods are efficient and superior to most existing robust methods for data reconstruction.

Linear Discriminant analysis (LDA) has been widely used for face recognition. However, when identifying faces in the wild, the existence of outliers that deviate significantly from the rest of data can arbitrarily skew the desired solution. This usually deteriorates LDA's performance dramatically, thus preventing it from mass deployment in real-world applications. To handle this problem, we propose an effective distance metric learning method based LDA, namely Euler LDA-L21 (e-LDA-L21). e-LDA-L21 is carried out in two stages, in which each image is mapped into a complex space by Euler transform in the first stage and the ℓ2,1-norm is adopted as the distance metric in the second stage. This not only reveals nonlinear features but also exploits the geometric structure of data. To solve e-LDA-L21 efficiently, we propose an iterative algorithm, which is a closed-form solution at each iteration with convergence guaranteed. Finally, we extend e-LDA-L21 to Euler 2DLDA-L21 (e-2DLDA-L21) which further exploits the spatial information embedded in image pixels. Experimental results on several face databases demonstrate its superiority over the state-of-the-art algorithms.