Compressed sensing studies linear recovery problems under structure assumptions. We introduce a new class of measurement operators, coined hierarchical measurement operators, and prove results guaranteeing the efficient, stable and robust recovery of hierarchically structured signals from such measurements. We derive bounds on their hierarchical restricted isometry properties based on the restricted isometry constants of their constituent matrices, generalizing and extending prior work on Kronecker-product measurements. As an exemplary application, we apply the theory to two communication scenarios. The fast and scalable HiHTP algorithm is shown to be suitable for solving these types of problems and its performance is evaluated numerically in terms of sparse signal recovery and block detection capability.

压缩感知在结构假设下研究线性恢复问题。我们引入了一类新的测量算子，创造了分层测量算子，并证明了保证从此类测量中有效、稳定和稳健地恢复分层结构信号的结果。我们根据其组成矩阵的受限等距常数推导出其分层受限等距属性的界限，概括和扩展先前关于克罗内克积测量的工作。作为示例应用，我们将该理论应用于两种通信场景。快速且可扩展的 HiHTP 算法被证明适用于解决这些类型的问题，并且在稀疏信号恢复和块检测能力方面对其性能进行了数值评估。