Hierarchically Block-Sparse Recovery With Prior Support Information

Abstract: We provide new recovery bounds for hierarchical compressed sensing (HCS) based on prior support information (PSI). A detailed PSI-enabled reconstruction model is formulated using various forms of PSI. The hierarchical block orthogonal matching pursuit with PSI (HiBOMP-P) algorithm is designed in a recursive form to reliably recover hierarchically block-sparse signals. We derive exact recovery conditions (ERCs) measured by the mutual incoherence property (MIP), wherein hierarchical MIP concepts are proposed, and further develop reconstructible sparsity levels to reveal sufficient conditions for ERCs. Leveraging these MIP analyses, we present several extended insights, including reliable recovery conditions in noisy scenarios and the optimal hierarchical structure for cases where sparsity is not equal to zero. Our results further confirm that HCS offers improved recovery performance even when the prior information does not overlap with the true support set, whereas existing methods heavily rely on this overlap, thereby compromising performance if it is absent.