Ratio sum formula for dimensionality reduction

High-dimensional data analysis often suffers the so-called curse of dimensionality. Therefore, dimensionality reduction is usually carried out on the high-dimensional data before the actual analysis, which is a common and efficient way to eliminate this effect. And the popular trace ratio criterion is an extension of the original linear discriminant analysis (LDA) problem, which involves a search of a transformation matrix W to embed high-dimensional space into a low-dimensional space to achieve dimensionality reduction. However, the trace ratio criterion tends to obtain projection direction with very small variance, which the subset after the projection is diffcult to present the most representative information of the data with maximum efficiency. In this paper, we target on this problem and propose the ratio sum formula for dimensionality reduction. Firstly, we analyze the impact of this trend. Then in order to solve this problem, we propose a new ratio sum formula as well as the solution. In the end, we perform experiments on the Yale-B, ORL, and COIL-20 data sets. The theoretical studies and actual numerical analysis confirm the effectiveness of the proposed method.