ANALYSIS VS SYNTHESIS - AN INVESTIGATION OF (CO)SPARSE SIGNAL MODELS ON GRAPHS

In this work, we present a theoretical study of signals with sparse representations in the vertex domain of a graph, which is primarily motivated by the discrepancy arising from respectively adopting a synthesis and analysis view of the graph Laplacian matrix. Sparsity on graphs and, in particular, the characterization of the subspaces of signals which are sparse with respect to the connectivity of the graph, as induced by analysis with a suitable graph operator, remains in general an opaque concept which we aim to elucidate. By leveraging the theory of cosparsity, we present a novel (co)sparse graph Laplacian-based signal model and characterize the underlying (structured) (co)sparsity, smoothness and localization of its solution subspaces on undirected graphs, while providing more refined statements for special cases such as circulant graphs. Ultimately, we substantiate fundamental discrepancies between the cosparse analysis and sparse synthesis models in this structured setting, by demonstrating that the former constitutes a special, constrained instance of the latter.

1. INTRODUCTION

The ability to capture geometric complexity, amid an ever-growing presence of large, irregularly and complex structured data, has rendered graphs a powerful tool for arising representation and processing tasks. Simultaneously, the extension of classical signal processing concepts and tools to signals defined on graphs has created a need for a comprehensive theoretical foundation, culminating in the formation of the field of Graph Signal Processing (GSP) [1]. With its underlying theory still in its infancy, new questions and challenges are continuously emerging as a result of the complex connectivity of networks.