Research On Dimension Reduction Methods Of High Dimensional Data

This paper presents an overview of the eigenvalue problems that are related to dimension reduction. Given some input high-dimensional data, the goal of dimension reduction is to map them to a low dimensional space such that certain properties of the initial data are preserved. Optimizing the above properties among the reduced data can be typically formulated as a trace optimization problem that leads to an eigenvalue problem. There is a rich variety of such problems and the goal of this paper is to unravel relations between them as well as to discuss effective solution techniques. First, we make a distinction between projective methods that determine an explicit linear projection from the high-dimensional space to the low-dimensional space. Then, we show that all of the eigenvalue problems solved in the context of explicit projections can be viewed as the projected analogues of the so-called nonlinear or implicit projections. Finally, by introducing a new distance measure formula to replace the Euclidean distance in the algorithm, an improved algorithm was proposed to overcome the shortcomings of the locally linear embedding algorithm that was not suitable for non-uniform distribution data and did not use the information of distant points. In the UCI data sets the experimental results show that dimensionality reduction results better than the original algorithm.