Sparse Locally Linear Embedding

The Locally Linear Embedding (LLE) algorithm has proven useful for determining structure preserving, dimension reducing mappings of data on manifolds. We propose a modification to the LLE optimization problem that serves to minimize the number of neighbors required for the representation of each data point. The algorithm is shown to be robust over wide ranges of the sparsity parameter producing an average number of nearest neighbors that is consistent with the best performing parameter selection for LLE. Given the number of non-zero weights may be substantially reduced in comparison to LLE, Sparse LLE can be applied to larger data sets. We provide three numerical examples including a color image, the standard swiss roll, and a gene expression data set to illustrate the behavior of the method in comparison to LLE. The resulting algorithm produces comparatively sparse representations that preserve the neighborhood geometry of the data in the spirit of LLE.