Local Neighborhood Embedding for Unsupervised Nonlinear Dimension Reduction

The construction of similarity relationship among data points plays a critical role in manifold learning. There exist two popular schemes, i.e., pairwise-distance based similarity and reconstruction coefficient based similarity. Existing works only have involved one scheme of them. These two schemes have different drawbacks. For pairwisedistance based similarity graph algorithms, they are sensitive to the noise and outliers. For reconstruction coefficient based similarity graph algorithms, they need sufficient sampled data and the neighborhood size is sensitive. This paper proposes a novel algorithm, called Local Neighborhood Embedding (LNE), which preserves pairwise-distance based similarity and reconstruction coefficient based similarity for finding the latent low dimensional structure of data. It has following three advantages: Firstly,it is insensitive to the choice of neighborhood size; Secondly, it is robust to the noise; Thirdly, It works well even in under-sampled case. Furthermore, the proposed objective function has a closedform solution, which means it has a low computational complexity, and the experimental results illustrate that LNE has a competitive performance in dimensionality reduction. Index terms Dimension reduction, manifold learning, similarity graph, unsupervised learning