Traditional Discriminant analysis (DA) is one of the classical supervised learning algorithms to reduce the dimensionality of data with Gaussian assumption. Since the unique class mean in traditional DA is intractable to estimate the non-Gaussian distrbution of data, some existing DA algorithms based on the clustering criterion focus on learning multiple means in each class so as to address the non-Gaussian issue. The clustering-based DA inevitably involved the constraint optimization problem to learn multiple means, which may lead to the locally optimal solution. To address these issues, inspired by the smooth approximation theory and the concept of Kolmogorov mean, this paper explores an unconstraint function with asymptotic property as an alternative proxy to clustering-based DA algorithms. Thus the derived DA algorithm, i.e., adaptive and asymptotic mean-based subclass discriminant analysis (AASDA), which not only leverages multiple means to represent different subclasses in same class but also adaptively and asymptotically learns the similar mean for each sample in the learned optimal subspace via the gradient-based optimizer. The asymptotic analysis of unconstraint function, the gradient analysis and convergence guarantee of proposed criterion verify the effectiveness of AASDA algorithm. Its merits are thoroughly assessed on a suite of synthetic and real world data experiments.

Principal component analysis (PCA) is one of the most versatile techniques for unsupervised dimension reduction, which is implemented as a fundamental preprocessing method in multiple tasks of statistics and machine learning research because of its efficiency. Nevertheless, researchers have concentrated on the identification of outliers that do not conform to the low-dimensional approximation through statistical methods, e.g., outlier rejection, without giving insights on each data point with a dynamic ratio of signal-to-noise components in the high-dimensional regimes. To characterize the dynamic nature of the principal component information, we propose a Normalized Robust PCA with Adaptive Reconstruction Error minimization model, which considers both the adaptive normalization technique and flexible weights learning simultaneously. With this configuration, the principal component information constantly adjusts the degree of sparsity for activated samples. In other words, the signal component's discrimination and noise information restriction could work cooperatively. Empirical studies on one synthetic dataset and several benchmarks demonstrate the effectiveness of our proposed method over existing outlier rejection methods.