Novel min-max reformulations of Linear Inverse Problems
This work addresses signal recovery from few measurements in applications like medical imaging, but it appears incremental as it builds on existing LIP frameworks with a new reformulation.
The authors tackled ill-posed Linear Inverse Problems (LIP) by proposing a generalized error-constrained version and deriving an equivalent convex-concave min-max reformulation, which enables simple saddle-point algorithms to find solutions and aids in dictionary learning with recovery constraints.
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.