A Spectral Proximal Methodforsparse Optimisation On Underdetermined Linear Systems - Woo, Gillian Yi Han (2022)
In this research, we will solve the l0-norm sparse optimisation problem. This is a l0-norm problem with an underdetermined system as its constraint. Using the Lagrangian method, this problem is transformed into an unconstrained optimisation problem. However, it cannot be solved by using the standard optimisation algorithm since l0-norm is nonconvex and non-smooth. Hence, a new method, the spectral proximal method (SPM) has been proposed and applied to the l0-norm unconstrained optimisation problem. This method is a combination of the proximal method and spectral gradient method. Based on previous research, the performance of the spectral gradient method is better than the other standard unconstrained optimisation methods due to the fact that the approximation of the full rank Hessian matrix is replaced by a diagonal matrix. Hence, the memory required O(n) storage instead of O(n2) storage. Convergent analysis of this method is established. The efficiency of the proposed method with the existing version of proximal gradient methods as its benchmarks are compared using Python software on simulated datasets and also large real-world MNIST datasets. The results show that our proposed method is more robust and stable for finding sparse solution of the linear system.