Identifiability in Blind Deconvolution under Minimal Assumptions

Blind deconvolution (BD) arises in many applications. Without assumptions on the signal and the filter, BD is ill-posed. 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 of ours, we provided the first algebraic sample complexities for BD with almost all bases or frames. We showed that for BD of vectors in $\mathbb{C}^n$, with two subspace constraints of dimensions $m_1$ and $m_2$, the required sample complexity is $n\geq m_1m_2$. This result is suboptimal, since the number of degrees of freedom is merely $m_1+m_2-1$. We provided analogus 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 low-rank matrix recovery, we are finally able to bridge this gap, and derive an optimal sample complexity result for BD with generic bases or frames. We show that for BD of vectors in $\mathbb{C}^n$, with two subspace constraints of dimensions $m_1$ and $m_2$, the required sample complexity is $n > m_1+m_2$. We present the extensions of this result to BD with sparsity constraints or mixed constraints, with the sparsity level replacing the subspace dimension.