  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.
