Adaptive multi-penalty regularization based on a generalized Lasso path

Abstract

For many algorithms, parameter tuning remains a challenging task, which becomes tedious in a multi-parameter setting. Multi-penalty regularization, successfully used for solving undetermined sparse regression problems of unmixing type, is one of such examples. We propose a novel algorithmic framework for an adaptive parameter choice in multi-penalty regularization with focus on correct support recovery. By extending ideas on regularization paths, we provide an efficient procedure for the construction of regions containing structurally similar solutions, i.e., solutions with the same sparsity and sign pattern, over the range of parameters. Combined with a model selection criterion, regularization parameters are chosen in a data-adaptive manner. Another advantage of our algorithm is that it provides an overview on the solution stability over the parameter range. We provide a numerical analysis of our method and compare it to the state-of-the-art algorithms for compressed sensing problems to demonstrate the robustness and power of the proposed algorithm.