Robust locally nonlinear embedding (RLNE) for dimensionality reduction of high-dimensional data with noise

Local Linear Embedding (LLE) is a nonlinear manifold learning method for dimensionality reduction in high-dimensional data. However, when the data is distorted by noise, efficiency of LLE significantly diminishes. This paper proposes a robust locally nonlinear embedding (RLNE) method to alleviate the impact of noise. This is achieved by constructing nonlinear functions between data neighbors in high-dimensional space, and then mapping the relationships to low manifolds. The constrained least squares method is used to obtain more uniform weights to ensure that the neighborhood is approximately located on the local nonlinear patches of the manifold. Theoretical analysis is conducted on the reasons underlying RLNE's robustness to noise. Experimental results on synthetic and real-world data highlight RLNE's ability to preserve the intrinsic structure of data, showcasing robustness across various types data with various levels of noise, as well as with a larger number of nearest neighbors.