On the impossibility of blind deconvolution for geometrically decaying subspace sparse signals

Blind deconvolution is an ubiquitous non-linear inverse problem in applications like wireless communications and image processing. This problem is generally ill-posed since signal identifiability is a key concern, and there have been efforts to use sparse models for regularizing blind deconvolution to promote signal identifiability. The identifiability of the sparse blind deconvolution problem is analyzed herein under assumptions of geometric decay of the channel and canonical sparsity of the source, and the ill-posedness is quantified by lower bounding the Hausdorff dimension of the unidentifiable signal set. The approach involves lifting the deconvolution problem into a rank one matrix recovery problem and analyzing the rank two null space of the resultant linear operator. An important conclusion of the paper is that the structural restriction of geometric decay of the channel vector with canonical sparsity of the source, occurring naturally in multi-hop channel estimation, is insufficient for signal identifiability in blind deconvolution, even if the canonical-sparse support is known.