Random Projection for Dimension Reduction and Mixture of Gaussians

• Main Authors: Meng-Hung Hsu, 許孟弘
• Other Authors: I-Ping Tu
• Format: Others
• Language: zh-TW
• Published: 2018
• Online Access: http://ndltd.ncl.edu.tw/handle/q2p6y4

碩士 === 國立臺灣大學 === 數學研究所 === 106 === Random projection is a promising dimensional reduction technique for high-dimensional data analysis. Johnson-Lindenstrauss Lemma states that a set of points in a high-dimensional space can be embedded into a space of lower dimension in such a way that distances between the points are nearly preserved. In other words, the structure of datasets is not destroyed by random projection. Besides mixtures of Gaussian are among the most fundamental and widely used statistical models. In this article, we show that when a mixtures Gaussians which are separated are projected, the projected Gaussians would be separated through random projection under some conditions. Moreover, the ratio of the eigenvalues of the covariance matrix of projected data becomes little compared with the ratio of the eigenvalues of the covariance matrix of original data. Finally, some numerical experiments with Gaussian mixtures will be illustrated.