  In this article, we dwell into the class of so-called ill-posed Linear
Inverse Problems (LIP) which simply refers to the task of recovering the entire
signal from its relatively few random linear measurements. Such problems arise
in a variety of settings with applications ranging from medical image
processing, recommender systems, etc. We propose a slightly generalized version
of the error constrained linear inverse problem and obtain a novel and
equivalent convex-concave min-max reformulation by providing an exposition to
its convex geometry. Saddle points of the min-max problem are completely
characterized in terms of a solution to the LIP, and vice versa. Applying
simple saddle point seeking ascend-descent type algorithms to solve the min-max
problems provides novel and simple algorithms to find a solution to the LIP.
Moreover, the reformulation of an LIP as the min-max problem provided in this
article is crucial in developing methods to solve the dictionary learning
problem with almost sure recovery constraints.
