2015/03/12 by Jingwei Liang, Liang, Jingwei, Jalal Fadili +3
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1503.03703
openalex publication_date 2015/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider a class of Forward--Backward (FB) splitting methods that includes several variants (e.g. inertial schemes, FISTA) for minimizing the sum of two proper convex and lower semi-continuous functions, one of which has a Lipschitz continuous gradient, and the other is partly smooth relatively to a smooth active manifold M. We propose a unified framework, under which we show that, this class of FB-type algorithms (i) correctly identifies the active manifolds in a finite number of iterations (finite activity identification), and (ii) then enters a local linear convergence regime, which we characterize precisely in terms of the structure of the underlying active manifolds. For simpler problems involving polyhedral functions, we show finite termination. We also establish and explain why FISTA (with convergent sequences) locally oscillates and can be slower than FB. These results may have numerous applications including in signal/image processing, sparse recovery and machine learning. Indeed, the obtained results explain the typical behaviour that has been observed numerically for many problems in these fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to name only a few.