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