Dimensionality Reduction Based on Multilocal Linear Pattern Preservation

Manifold learning-based methods, such as LLE, capture the geometry of the data based on the assumption that the local structure of a manifold is linear. However, these methods may extract an inaccurate local structure when the nonlinearity of the data is obvious. In this paper, we propose a novel dimensionality reduction method with the ability to characterize the locally nonlinear geometry of the data by multilocal linearity. Specifically, we first construct a local area for each data point. And based on the overlapping of local areas, each data point will belong to and be linearly reconstructed from several local areas. Next, the set of linear coefficients used to reconstruct the data point constitutes the multilocal linear pattern (MLLP) which is used to characterize the local geometry of the data. Geometrically, the MLLP of a data point represents the hyperplanes in different directions passing through the current point. And the locally nonlinear surface where the data point is located is approximated by these hyperplanes, which is more accurate to reflect the geometry of the data. Then, MLLP is preserved to the embedding data space, and the dimension-reduced data can be obtained by minimizing the reconstruction errors. Finally, experiment results on various datasets demonstrate the effectiveness of the proposed method.