Linear dimensionality reduction method based on topological properties

Abstract

Dimensionality reduction is an important data preprocessing technique that has been extensively studied in machine learning and data mining. Locality Preserving Projection (LPP) is a widely used linear unsupervised dimensionality reduction method, which maps high-dimensional data into low-dimensional subspace through linear transformation. Although various variants of LPP have been proposed to tackle different drawbacks of LPP, it is identified in this article that LPP does not possess the important topological property of translation invariance, that is, the linear transformation given by LPP is strongly related to the relative position between the data and the origin of the coordinate system. In this article, we theoretically analyze the reason why this drawback exists in LPP and propose to resolve it by introducing a kind of centralization to the model. Moreover, as topological properties are prominent information to characterize the structure of the data, this article proposes a further improvement of LPP to maintain topological connectivity of data after dimensionality reduction. Experiments on multiple synthetic and real-world datasets show that the new model incorporating topological properties outperforms not only the original LPP model but also several other classic linear or non-linear dimensionality reduction methods.