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

Samples from intimate (non-linear) mixtures are generally modeled as being drawn from a smooth manifold. Scenarios where the data contains multiple intimate mixtures with some constituent materials in common can be thought of as manifolds which share a boundary. Two important steps in the processing of such data are (i) to identify (cluster) the different mixture-manifolds present in the data and (ii) to eliminate the non-linearities present the data by mapping each mixture-manifold into some low-dimensional euclidean space (embedding). Manifold clustering and embedding techniques appear to be an ideal tool for this task, but the present state-of-the-art algorithms perform poorly for hyperspectral data, particularly in the embedding task. We propose a novel reconstruction-based algorithm for improved clustering and embedding of mixture-manifolds. The algorithms 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 neighbors on the same manifold as the target-point are used in the reconstruction. The reconstruction matrix generated by using this technique is block-diagonal and can be used for clustering (using spectral clustering) and embedding. The improved performance of the algorithms vis-a-vis its competitors is exhibited on a variety of simulated and real mixture datasets.