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In this article, we dwell into the class of so-called ill-posed LinearInverse Problems (LIP) which simply refers to the task of recovering the entiresignal from its relatively few random linear measurements. Such problems arisein a variety of settings with applications ranging from medical imageprocessing, recommender systems, etc. We propose a slightly generalized versionof the error constrained linear inverse problem and obtain a novel andequivalent convex-concave min-max reformulation by providing an exposition toits convex geometry. Saddle points of the min-max problem are completelycharacterized in terms of a solution to the LIP, and vice versa. Applyingsimple saddle point seeking ascend-descent type algorithms to solve the min-maxproblems 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 thisarticle is crucial in developing methods to solve the dictionary learningproblem with almost sure recovery constraints.

Novel min-max reformulations of Linear Inverse Problems