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In this work, we present a theoretical study of signals with sparserepresentations in the vertex domain of a graph, which is primarily motivatedby the discrepancy arising from respectively adopting a synthesis and analysisview of the graph Laplacian matrix. Sparsity on graphs and, in particular, thecharacterization of the subspaces of signals which are sparse with respect tothe connectivity of the graph, as induced by analysis with a suitable graphoperator, remains in general an opaque concept which we aim to elucidate. Byleveraging the theory of cosparsity, we present a novel (co)sparse graphLaplacian-based signal model and characterize the underlying (structured)(co)sparsity, smoothness and localization of its solution subspaces onundirected graphs, while providing more refined statements for special casessuch as circulant graphs. Ultimately, we substantiate fundamental discrepanciesbetween the cosparse analysis and sparse synthesis models in this structuredsetting, by demonstrating that the former constitutes a special, constrainedinstance of the latter.

Analysis vs Synthesis - An Investigation of (Co)sparse Signal Models on Graphs