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.

在过去的十年中，稀疏表示模型已在众多信号和图像处理应用程序中取得了显著成果。为了合并数据的固有结构并考虑并非所有支持模式都具有同等可能性的事实，通过强制执行各种结构稀疏模式来丰富了该模型。图规则化稀疏编码是一种可能的经典稀疏编码扩展，它是由图形信号处理的出现引起的。该模型明确考虑了数据域的固有几何结构，并已成功应用于各种应用中。但是，重点是开发算法解决方案，迄今为止，该问题的理论基础一直落后。在这项工作中 我们填补了这一空白，并提出了图形正则化稀疏编码问题的新颖理论分析，为获得的解决方案的稳定性以及几种追踪技术的成功提供了最坏的保证。此外，我们制定了建立图正则化稀疏编码解决方案优于结构不可知稀疏编码对应物的优势的条件。