L1Gss

L1Gss stands for L1 Generalized Sparse Solver, a family of optimization algorithms designed to solve large-scale L1-regularized problems by exploiting sparsity. It extends conventional proximal gradient and ADMM approaches by employing a generalized shrinkage operator capable of enforcing various sparsity structures, including block and overlapping group sparsity. The framework is used in signal processing, statistics, and machine learning to recover sparse signals from limited or noisy measurements.

Originating in the early 2010s as researchers sought scalable methods for high-dimensional sparse recovery, L1Gss was described in several theoretical papers as an iteration scheme that minimizes F(x) = f(x) + lambda ||x||1, with f convex and differentiable. Its core innovation is the generalized shrinkage step S_lambda^g, which allows different sparsity patterns to be promoted without sacrificing convergence guarantees when paired with a suitable step size.

Algorithm and variants: Basic L1Gss iterates x^{k+1} = S_lambda^g(x^k - t_k grad f(x^k)). With additional splitting or dual variables, it can be implemented as an ADMM-based variant or in parallel, enabling efficient scaling to datasets with millions of features. Convergence is established under standard assumptions on f and proper selection of lambda and step size.

Applications and reception: L1Gss is used for sparse signal reconstruction in compressed sensing, feature selection in high-dimensional datasets, image and video denoising, and genomic data analysis. It is valued for robustness to noise, scalability, and its ability to incorporate structured sparsity. Limitations include sensitivity to regularization parameter choice and reduced performance on dense, non-sparse problems.

See also: LASSO, proximal gradient methods, ADMM, sparse optimization, structured sparsity.