Ratio sum formula for dimensionality reduction
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Ratio sum formula for dimensionality reduction Ke Liang1 · XiaoJun Yang1,2 · YuXiong Xu1 · Rong Wang3 · Feiping Nie3 Received: 9 March 2020 / Revised: 27 July 2020 / Accepted: 28 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords Ratio sum · Trace ratio · Feature selection · Dimensionality reduction
1 Introduction With the rapidly increasing size of the data, the high dimensional data appears frequently in many real applications of data mining, machine learning, and others [2, 28, 29]. Directly dealing with such high dimensional data is challenging since its computationally inefficient. Therefore, it is great practical significance to find the low-dimensional representation of high-dimensional space by dimension reduction method [11, 15, 27, 32, 35].
XiaoJun Yang
[email protected] 1
School of Information Engineering, Guangdong University of Technology, Guangzhou, Guangdong 510006, China
2
Synergy Innovation Institute of GDUT, Heyuan, Guangdong 517000, China
3
School of Computer Science and the Center for Optical Imagery Analysis and Learning, Northwestern Polytechnical University, Xi’an, 710072, China
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In recent decades, the dimensionality reduction method has been sufficiently developed to handle different types of high-dimensional data, such as images, texts, and videos [18, 31, 33]. One is based on the method of feature selection, which selects a subset of the original features according to some criteria [1]. Das [8], Danubianu et al. [7] and Nie et al. [21] describes the three most common feature selection methods, including filter methods, wrapper methods, and embedded methods. A study [19] shows an efficient semi-supervised feature selection algorithm to select relevant features using both labeled and unlabeled data. In another study [36], a nov
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