Parametric dictionary learning for topological signal representation

The aim of this paper is to introduce a novel dictionary learning algorithm for sparse representation of signals defined over regular cell complexes. Leveraging tools from Hodge theory, we inject the underlying topology in the dictionary structure by parametrizing it as a concatenation of sub-dictionaries that are polynomial of Hodge Laplacians. The learning problem is cast as the joint optimization of the topological dictionary coefficients and the sparse signal representation, which is efficiently solved via an iterative alternating algorithm. Numerical results on synthetic data show the effectiveness of the proposed procedure in learning sparse representations of topological signals.