Fast blind deconvolution and blind demixing via nonconvex optimization

We study the question of reconstructing a sequence of {f
_i
, g
_i
}
_i=1
^s
from the sum of their convolution, i.e., y = Σ
_i=1
^s
f
_i
* g
_i
. This problem is closely related to both blind deconvolution and blind demixing problem. Our goal is to find all {f
_i
, g
_i
}
_i=1
^s
by jointly demixing each component f
_i
* g
_i
and performing deconvolution procedure. While the convex program is able to solve this problem effectively and robustly under certain conditions, the main obstacle towards real-time deployment is to find a provably convergent, efficient and robust algorithm. We present an efficient numerical algorithm which guarantees the exact recovery of the solutions despite its notorious non-convexity. The proposed two-step algorithm converges to the global minima linearly and is also robust to the noise. Though the derived performance bound is suboptimal in terms of the information-theoretic limit, numerical simulations show the remarkable performance even if the number of measurements is close to the degree of freedom.