Multi-View Clustering via Semi-non-negative Tensor Factorization

Multi-view clustering (MVC) based on non-negative matrix factorization (NMF) and its variants have received a huge amount of attention in recent years due to their advantages in clustering interpretability. However, existing NMF-based multi-view clustering methods perform NMF on each view data respectively and ignore the impact of between-view. Thus, they can't well exploit the within-view spatial structure and between-view complementary information. To resolve this issue, we present semi-non-negative tensor factorization (Semi-NTF) and develop a novel multi-view clustering based on Semi-NTF with one-side orthogonal constraint. Our model directly performs Semi-NTF on the 3rd-order tensor which is composed of anchor graphs of views. Thus, our model directly considers the between-view relationship. Moreover, we use the tensor Schatten p-norm regularization as a rank approximation of the 3rd-order tensor which characterizes the cluster structure of multi-view data and exploits the between-view complementary information. In addition, we provide an optimization algorithm for the proposed method and prove mathematically that the algorithm always converges to the stationary KKT point. Extensive experiments on various benchmark datasets indicate that our proposed method is able to achieve satisfactory clustering performance

Discriminant Analysis via Joint Euler Transform and ℓ2, 1-Norm

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 the 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

Two-Dimensional PCA with F-Norm Minimization

Two-dimensional principle component analysis (2DPCA) has been widely used for face image representation and recognition. But it is sensitive to the presence of outliers. To alleviate this problem, we propose a novel robust 2DPCA, namely 2DPCA with F-norm minimization (F-2DPCA), which is intuitive and directly derived from 2DPCA. In F-2DPCA, distance in spatial dimensions (attribute dimensions) is measured in F-norm, while the summation over different data points uses 1-norm. Thus it is robust to outliers and rotational invariant as well. To solve F-2DPCA, we propose a fast iterative algorithm, which has a closed-form solution in each iteration, and prove its convergence. Experimental results on face image databases illustrate its effectiveness and advantages

Adaptive robust principal component analysis

Robust Principal Component Analysis (RPCA) is a powerful tool in machine learning and data mining problems. However, in many real-world applications, RPCA is unable to well encode the intrinsic geometric structure of data, thereby failing to obtain the lowest rank representation from the corrupted data. To cope with this problem, most existing methods impose the smooth manifold, which is artificially constructed by the original data. This reduces the flexibility of algorithms. Moreover, the graph, which is artificially constructed by the corrupted data, is inexact and does not characterize the true intrinsic structure of real data. To tackle this problem, we propose an adaptive RPCA (ARPCA) to recover the clean data from the high-dimensional corrupted data. Our proposed model is advantageous due to: 1) The graph is adaptively constructed upon the clean data such that the system is more flexible. 2) Our model simultaneously learns both clean data and similarity matrix that determines the construction of graph. 3) The clean data has the lowest-rank structure that enforces to correct the corruptions. Extensive experiments on several datasets illustrate the effectiveness of our model for clustering and low-rank recovery tasks