Haiyan Jiang

Haiyan Jiang, Shanshan Qin, Oscar Hernan Madrid Padilla

In this paper, we consider a new variant for principal component analysis (PCA), aiming to capture the grouping and/or sparse structures of factor loadings simultaneously. To achieve these goals, we employ a non-convex truncated regularization with naturally adjustable sparsity and grouping effects, and propose the Feature Grouping and Sparse Principal Component Analysis (FGSPCA). The proposed FGSPCA method encourages the factor loadings with similar values to collapse into disjoint homogeneous groups for feature grouping or into a special zero-valued group for feature selection, which in turn helps reducing model complexity and increasing model interpretation. Usually, existing structured PCA methods require prior knowledge to construct the regularization term. However, the proposed FGSPCA can simultaneously capture the grouping and/or sparse structures of factor loadings without any prior information. To solve the resulting non-convex optimization problem, we propose an alternating algorithm that incorporates the difference-of-convex programming, augmented Lagrange method and coordinate descent method. Experimental results demonstrate the promising performance and efficiency of the new method on both synthetic and real-world datasets. An R implementation of FGSPCA can be found on github {https://github.com/higeeks/FGSPCA}.

Haiyan Jiang, Haoyi Xiong, Dongrui Wu et al.

Principal component analysis (PCA) has been widely used as an effective technique for feature extraction and dimension reduction. In the High Dimension Low Sample Size (HDLSS) setting, one may prefer modified principal components, with penalized loadings, and automated penalty selection by implementing model selection among these different models with varying penalties. The earlier work [1, 2] has proposed penalized PCA, indicating the feasibility of model selection in $L_2$- penalized PCA through the solution path of Ridge regression, however, it is extremely time-consuming because of the intensive calculation of matrix inverse. In this paper, we propose a fast model selection method for penalized PCA, named Approximated Gradient Flow (AgFlow), which lowers the computation complexity through incorporating the implicit regularization effect introduced by (stochastic) gradient flow [3, 4] and obtains the complete solution path of $L_2$-penalized PCA under varying $L_2$-regularization. We perform extensive experiments on real-world datasets. AgFlow outperforms existing methods (Oja [5], Power [6], and Shamir [7] and the vanilla Ridge estimators) in terms of computation costs.

Haiyan Jiang, Shuyu Li, Luwei Zhang et al.

Although many techniques have been applied to matrix factorization (MF), they may not fully exploit the feature structure. In this paper, we incorporate the grouping effect into MF and propose a novel method called Robust Matrix Factorization with Grouping effect (GRMF). The grouping effect is a generalization of the sparsity effect, which conducts denoising by clustering similar values around multiple centers instead of just around 0. Compared with existing algorithms, the proposed GRMF can automatically learn the grouping structure and sparsity in MF without prior knowledge, by introducing a naturally adjustable non-convex regularization to achieve simultaneous sparsity and grouping effect. Specifically, GRMF uses an efficient alternating minimization framework to perform MF, in which the original non-convex problem is first converted into a convex problem through Difference-of-Convex (DC) programming, and then solved by Alternating Direction Method of Multipliers (ADMM). In addition, GRMF can be easily extended to the Non-negative Matrix Factorization (NMF) settings. Extensive experiments have been conducted using real-world data sets with outliers and contaminated noise, where the experimental results show that GRMF has promoted performance and robustness, compared to five benchmark algorithms.

Zhenliang Zhang, Dongdong Weng, Haiyan Jiang et al.

We propose a framework called inverse augmented reality (IAR) which describes the scenario that a virtual agent living in the virtual world can observe both virtual objects and real objects. This is different from the traditional augmented reality. The traditional virtual reality, mixed reality and augmented reality are all generated for humans, i.e., they are human-centered frameworks. On the contrary, the proposed inverse augmented reality is a virtual agent-centered framework, which represents and analyzes the reality from a virtual agent's perspective. In this paper, we elaborate the framework of inverse augmented reality to argue the equivalence of the virtual world and the physical world regarding the whole physical structure.