Trace ratio criterion based generalized discriminative learning for semi-supervised dimensionality reduction

Dealing with high-dimensional data has always been a major problem in many pattern recognition and machine learning applications. Trace ratio criterion is a criterion that can be applicable to many dimensionality reduction methods as it directly reflects Euclidean distance between data points of within or between classes. In this paper, we analyze the trace ratio problem and propose a new efficient algorithm to find the optimal solution. Based on the proposed algorithm, we are able to derive an orthogonal constrained semi-supervised learning framework. The new algorithm incorporates unlabeled data into training procedure so that it is able to preserve the discriminative structure as well as geometrical structure embedded in the original dataset. Under such a framework, many existing semi-supervised dimensionality reduction methods such as SDA, Lap-LDA, SSDR, SSMMC, can be improved using our proposed framework, which can also be used to formulate a corresponding kernel framework for handling nonlinear problems. Theoretical analysis indicates that there are certain relationships between linear and nonlinear methods. Finally, extensive simulations on synthetic dataset and real world dataset are presented to show the effectiveness of our algorithms. The results demonstrate that our proposed algorithm can achieve great superiority to other state-of-art algorithms.