Theoretical guarantees for graph sparse coding

Over the last decade, the sparse representation model has led to remarkable results in numerous signal and image processing applications. To incorporate the inherent structure of the data and account for the fact that not all support patterns are equally likely, this model was enriched by enforcing various structural sparsity patterns. One plausible such extension of classic sparse coding, instigated by the emergence of graph signal processing, is graph regularized sparse coding. This model explicitly considers the intrinsic geometrical structure of the data domain, and has been successfully employed in various applications. However, emphasis was given to developing algorithmic solutions, and to date, the theoretical foundations to this problem have been lagging behind. In this work, we fill this gap and present a novel theoretical analysis of the graph regularized sparse coding problem, providing worst-case guarantees for the stability of the obtained solution, as well as for the success of several pursuit techniques. Furthermore, we formulate the conditions for which the superiority of the graph regularized sparse coding solution over the structure-agnostic sparse coding counterpart is established.