Identifiability and Stability in Blind Deconvolution Under Minimal Assumptions

Blind deconvolution (BD) arises in many applications. Without assumptions on the signal and the filter, BD does not admit a unique solution. In practice, subspace or sparsity assumptions have shown the ability to reduce the search space and yield the unique solution. However, existing theoretical analysis on uniqueness in BD is rather limited. In an earlier paper, we provided the first algebraic sample complexities for BD that hold for Lebesgue almost all bases or frames. We showed that for BD of a pair of vectors in ℂⁿ, with subspace constraints of dimensions m₁ and m₂, respectively, a sample complexity of n ≥ m₁m₂ is sufficient. This result is suboptimal, since the number of degrees of freedom is merely m₁ + m₂ - 1. We provided analogous results, with similar suboptimality, for BD with sparsity or mixed subspace and sparsity constraints. In this paper, taking advantage of the recent progress on the information-theoretic limits of unique low-rank matrix recovery, we finally bridge this gap, and derive an optimal sample complexity result for BD with generic bases or frames. We show that for BD of an arbitrary pair (respectively, all pairs) of vectors in ℂⁿ, with sparsity constraints of sparsity levels s1 and s2, a sample complexity of n > s₁ + s₂ [respectively, n > 2(s₁+s₂)] is sufficient. We also present analogous results for BD with subspace constraints or mixed constraints, with the subspace dimension replacing the sparsity level. Last but not least, in all the above scenarios, if the bases or frames follow a probabilistic distribution specified in this paper, the recovery is not only unique, but also stable against small perturbations in the measurements, under the same sample complexities.