Manifold Learning by Curved Cosine Mapping

Abstract

In the field of pattern recognition, data analysis, and machine learning, data points are usually modeled as high-dimensional vectors. Due to the curse-of-dimensionality, it is non-trivial to efficiently process the orginal data directly. Given the unique properties of nonlinear dimensionality reduction techniques, nonlinear learning methods are widely adopted to reduce the dimension of data. However, existing nonlinear learning methods fail in many real applications because of the too-strict requirements (for real data) or the difficulty in parameters tuning. Therefore, in this paper, we investigate the manifold learning methods which belong to the family of nonlinear dimensionality reduction methods. Specifically, we proposed a new manifold learning principle for dimensionality reduction named Curved Cosine Mapping (CCM). Based on the law of cosines in Euclidean space, CCM applies a brand new mapping pattern to manifold learning. In CCM, the nonlinear geometric relationships are obtained by utlizing the law of cosines, and then quantified as the dimensionality-reduced features. Compared with the existing approaches, the model has weaker theoretical assumptions over the input data. Moreover, to further reduce the computation cost, an optimized version of CCM is developed. Finally, we conduct extensive experiments over both artificial and real-world datasets to demonstrate the performance of proposed techniques.