  We consider the problem of demixing a sequence of source signals from the sum
of noisy bilinear measurements. It is a generalized mathematical model for
blind demixing with blind deconvolution, which is prevalent across the areas of
dictionary learning, image processing, and communications. However, state-of-
the-art convex methods for blind demixing via semidefinite programming are
computationally infeasible for large-scale problems. Although the existing
nonconvex algorithms are able to address the scaling issue, they normally
require proper regularization to establish optimality guarantees. The
additional regularization yields tedious algorithmic parameters and pessimistic
convergence rates with conservative step sizes. To address the limitations of
existing methods, we thus develop a provable nonconvex demixing procedure
viaWirtinger flow, much like vanilla gradient descent, to harness the benefits
of regularization-free fast convergence rate with aggressive step size and
computational optimality guarantees. This is achieved by exploiting the benign
geometry of the blind demixing problem, thereby revealing that Wirtinger flow
enforces the regularization-free iterates in the region of strong convexity and
qualified level of smoothness, where the step size can be chosen aggressively.
