Stability in blind deconvolution of sparse signals and reconstruction by alternating minimization

Blind deconvolution is the recovery of two unknown signals from their convolution. To overcome the ill-posedness of this problem, solutions based on priors tailored to specific application have been developed in practical applications. In particular, sparsity models have provided promising priors. In spite of empirical success in many applications, existing analyses are rather limited in two main ways: by disparity between theoretical assumptions on the signal and/or measurement model versus practical setups; or by failure to demonstrate success for parameter values within the optimal regime defined by the information theoretic limits. In particular, it has been shown that a naive sparsity model is not strong enough as a prior for identifiability in blind deconvolution problem. In addition to sparsity, we adopt a conic constraint by Ahmed et al., which enforces flat spectra in the Fourier domain. Under this prior, we provide an iterative algorithm that achieves guaranteed performance in blind deconvolution with number of measurements proportional (up to a logarithmic factor) to the sparsity level of the signal.