Weighed l1 on the simplex: Compressive sensing meets locality

TL;DR

This paper introduces weighted  and  metrics tailored for dictionary-based manifold learning, addressing geometric data structures and local geometry, with theoretical equivalence and practical optimization methods.

Contribution

It proposes novel weighted  and  regularizations for manifold learning that account for local geometry and data structure, with theoretical and empirical validation.

Findings

Weighted  and  are equivalent under Delaunay triangulation assumptions.

The proposed regularization improves dictionary learning for manifold data.

Optimization methods effectively learn dictionaries and sparse codes on synthetic and real data.

Abstract

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 $\ell_1$ minimization problem. We propose weighted $\ell_0$ and weighted $\ell_1$ metrics 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 $\ell_0$ and weighted $\ell_1$. We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.