Weighed ℓ1 on the Simplex: Compressive Sensing Meets Locality

Sparse manifold learning algorithms combine techniques in manifold learning and sparse optimization to learn features that could be utilized for downstream tasks. The standard setting of compressive sensing can not be immediately applied to this setup. Due to the intrinsic geometric structure of data, dictionary atoms might be redundant and do not satisfy the restricted isometry property or coherence condition. In addition, manifold learning emphasizes learning local geometry which is not reflected in a standard ℓ1 minimization problem. We propose weighted ℓ0 and weighted ℓ1 regularizations that encourage representation via neighborhood atoms suited for dictionary based manifold learning. Assuming that the data is generated from Delaunay triangulation, we show the equivalence of weighted ℓ1 and weighted ℓ0. We discuss an optimization program that learns the dictionaries and sparse co-efficients and demonstrate the utility of our regularization on synthetic and real datasets.