On Convex Duality in Linear Inverse Problems
This work addresses signal recovery challenges in applications like medical imaging and recommender systems, but appears incremental as it builds on existing convex duality theory.
The paper tackles the recovery of signals from few linear measurements in ill-posed Linear Inverse Problems (LIPs) by introducing a convex-concave min-max reformulation, which enables simple ascend-descent algorithms 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) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from its relatively few random linear measurements. Such problems arise in variety of settings with applications ranging from medical image processing, recommender systems etc. We provide an exposition to the convex duality of the linear inverse problems, and obtain a novel and equivalent convex-concave min-max reformulation that gives rise to simple ascend-descent type algorithms to solve an LIP. Moreover, such a reformulation is crucial in developing methods to solve the dictionary learning problem with almost sure recovery constraints.