Lei‐Hong Zhang

Lei-Hong Zhang, Ya-Nan Zhang

In this paper, we propose a novel dual-based Lawson's method, termed {b-d-Lawson}, designed for addressing the rational minimax approximation under specific interpolation conditions. The {b-d-Lawson} approach incorporates two pivotal components that have been recently gained prominence in the realm of the rational approximations: the barycentric representation of the rational function and the dual framework for tackling minimax approximation challenges. The employment of barycentric formulae enables a streamlined parameterization of the rational function, ensuring natural satisfaction of interpolation conditions while mitigating numerical instability typically associated with Vandermonde basis matrices when monomial bases are utilized. This enhances both the accuracy and computational stability of the method. To address the bi-level min-max structure, the dual framework effectively transforms the challenge into a max-min dual problem, thereby facilitating the efficient application of Lawson's iteration. The integration of this dual perspective is crucial for optimizing the approximation process. We will discuss several applications of interpolation-constrained rational minimax approximation and illustrate numerical results to evaluate the performance of the {b-d-Lawson} method.

Li Wang, Lei-Hong Zhang, Ren-Cang Li

A trace ratio optimization problem over the Stiefel manifold is investigated from the perspectives of both theory and numerical computations. At least three special cases of the problem have arisen from Fisher linear discriminant analysis, canonical correlation analysis, and unbalanced Procrustes problem, respectively. Necessary conditions in the form of nonlinear eigenvalue problem with eigenvector dependency are established and a numerical method based on the self-consistent field (SCF) iteration is designed and proved to be always convergent. As an application to multi-view subspace learning, a new framework and its instantiated concrete models are proposed and demonstrated on real world data sets. Numerical results show that the efficiency of the proposed numerical methods and effectiveness of the new multi-view subspace learning models.

Li Wang, Lei-Hong Zhang, Chungen Shen et al.

Multi-view datasets are increasingly collected in many real-world applications, and we have seen better learning performance by existing multi-view learning methods than by conventional single-view learning methods applied to each view individually. But, most of these multi-view learning methods are built on the assumption that at each instance no view is missing and all data points from all views must be perfectly paired. Hence they cannot handle unpaired data but ignore them completely from their learning process. However, unpaired data can be more abundant in reality than paired ones and simply ignoring all unpaired data incur tremendous waste in resources. In this paper, we focus on learning uncorrelated features by semi-paired subspace learning, motivated by many existing works that show great successes of learning uncorrelated features. Specifically, we propose a generalized uncorrelated multi-view subspace learning framework, which can naturally integrate many proven learning criteria on the semi-paired data. To showcase the flexibility of the framework, we instantiate five new semi-paired models for both unsupervised and semi-supervised learning. We also design a successive alternating approximation (SAA) method to solve the resulting optimization problem and the method can be combined with the powerful Krylov subspace projection technique if needed. Extensive experimental results on multi-view feature extraction and multi-modality classification show that our proposed models perform competitively to or better than the baselines.

Li Wang, Leihong Zhang, Chungen Shen et al.

We propose a unified framework for multi-view subspace learning to learn individual orthogonal projections for all views. The framework integrates the correlations within multiple views, supervised discriminant capacity, and distance preservation in a concise and compact way. It not only includes several existing models as special cases, but also inspires new novel models. To demonstrate its versatility to handle different learning scenarios, we showcase three new multi-view discriminant analysis models and two new multi-view multi-label classification ones under this framework. An efficient numerical method based on successive approximations via eigenvectors is presented to solve the associated optimization problem. The method is built upon an iterative Krylov subspace method which can easily scale up for high-dimensional datasets. Extensive experiments are conducted on various real-world datasets for multi-view discriminant analysis and multi-view multi-label classification. The experimental results demonstrate that the proposed models are consistently competitive to and often better than the compared methods that do not learn orthogonal projections.

Leihong Zhang, Li Wang, Zhaojun Bai et al.

We propose an efficient algorithm for solving orthogonal canonical correlation analysis (OCCA) in the form of trace-fractional structure and orthogonal linear projections. Even though orthogonality has been widely used and proved to be a useful criterion for pattern recognition and feature extraction, existing methods for solving OCCA problem are either numerical unstable by relying on a deflation scheme, or less efficient by directly using generic optimization methods. In this paper, we propose an alternating numerical scheme whose core is the sub-maximization problem in the trace-fractional form with an orthogonal constraint. A customized self-consistent-field (SCF) iteration for this sub-maximization problem is devised. It is proved that the SCF iteration is globally convergent to a KKT point and that the alternating numerical scheme always converges. We further formulate a new trace-fractional maximization problem for orthogonal multiset CCA (OMCCA) and then propose an efficient algorithm with an either Jacobi-style or Gauss-Seidel-style updating scheme based on the same SCF iteration. Extensive experiments are conducted to evaluate the proposed algorithms against existing methods including two real world applications: multi-label classification and multi-view feature extraction. Experimental results show that our methods not only perform competitively to or better than baselines but also are more efficient.