A novel approach is developed for nonlinear compression and reconstruction of
high-dimensional data vectors living on a smooth but otherwise unknown
manifold. Compression is effected through locally affine embeddings to
lower-dimensional spaces. These embeddings are obtained via dictionary learning
algorithms that leverage manifold smoothness as well as sparsity of the affine
model and its residuals. The emergent unifying framework is general enough to
encompass known locally linear embedding and compressive sampling approaches to
dimensionality reduction. Emphasis is placed on reconstructing high-dimensional
vectors from their low-dimensional
embeddings. Preliminary tests demonstrate the analytical claims, and their
potential to (de)compressing data and video sources.