A Dynamic Neighborhood Selection Approach for Locally Linear Embedding

Locally linear embedding is based on the assumption that the whole data manifolds are evenly distributed so that they determine the neighborhood for all points with the same neighborhood size. Accordingly, they fail to nicely deal with most real problems that are unevenly distributed. This paper presents a new approach that takes the general conceptual framework of Hessian locally linear embedding so as to guarantee its correctness in the setting of local isometry to an open connected subset but dynamically determines the local neighborhood size for each point. This approach estimates the approximate geodesic distance between any two points by the shortest path in the local neighborhood graph, and then determines the neighborhood size for each point by using the relationship between its local estimated geodesic distance matrix and local Euclidean distance matrix. This approach has clear geometry intuition as well as the better performance and stability to deal with the sparsely sampled or noise contaminated data sets that are often unevenly distributed. The conducted experiments on benchmark data sets validate the proposed approach.

PROJECTIVE NOISE CLEANING WITH DYNAMIC NEIGHBORHOOD SELECTION

In recent years, several methods of noise cleaning have been devised, of which projective methods have been particularly effective. In our paper, we explain in detail why orthogonal projections are nonoptimal and how the nonorthogonal projections suggested by Grassberger et al., naturally emerge from the SVD method. We show that this approach when combined with a dynamic neighborhood selection yields optimal results of noise cleaning.