On Clustering and Embedding Mixture Manifolds Using a Low Rank Neighborhood Approach

Spectra from a single intimate (nonlinear) mixture can be modeled as data points drawn from a smooth manifold. Spectral data sets containing hyperspectral observations of multiple intimate mixtures with some constituent materials in common can, therefore, be modeled as data clouds, in which each point is drawn from a union of manifolds that share a boundary. Two important steps in the processing of such data are to: 1) identify the different mixture manifolds present in the data and 2) invert the nonlinear mixing function by mapping each mixture manifold into some low-dimensional Euclidean space (manifold embedding). The present state-of-the-art algorithms for joint manifold clustering and embedding perform poorly for hyperspectral data, particularly in the embedding task. We propose a novel reconstruction-based algorithm for the improved clustering and the embedding of mixture manifolds. The algorithm attempts to reconstruct each target point as an affine combination of its nearest neighbors with an additional rank penalty on the neighborhood to ensure that only the neighbors on the same manifold as the target point are used in the reconstruction. The reconstruction matrix generated by this technique is both block diagonal and neighborhood-based, leading to improved clustering and embedding. The improved performance of the algorithm against its competitors is exhibited on a variety of simulated and real mixture data sets.