Hierarchical Sparse Recovery from Hierarchically Structured Measurements with Application to Massive Random Access

A new family of operators, dubbed hierarchical measurement operators, is introduced and discussed within the framework of hierarchically sparse recovery. A hierarchical measurement operator is a composition of block and mixing operations. It notably contains Kronecker products as a special case. Results on their hierarchical restricted isometry property (HiRIP) are derived, generalizing prior work on the recovery of hierarchically sparse signals from Kronecker-structured linear measurements. Specifically, these results show that recovery properties of the block and mixing part can be traded against each other. The measurement structure is motivated by a massive random access channel design in communication engineering. Numerical evaluation of user detection rates demonstrate benefits of the theoretical framework.