Relaxed Analysis Operator Learning

The problem of analysis operator learning can be formulated as a constrained optimisation problem. This problem has been approximately solved using projected gradient or geometric gradient descent methods. We will propose a relaxation for the constrained analysis operator learning in this paper. The relaxation has been suggested here to, a) reduce the computational complexity of the optimisation and b) include a larger set of admissible operators. We will show here that an appropriate relaxation can be useful in presenting a projection-free optimisation algorithm, while preventing the problem to become ill-posed. The relaxed optimisation objective is not convex and it is thus not always possible to find the global optimum. However, when a rich set of training samples are given, we empirically show that the desired synthetic analysis operator is recoverable, using the introduced sub-gradient descent algorithm.