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$ \ell_1 $-regularized linear inverse problems are frequently used in signalprocessing, image analysis, and statistics. The correct choice of theregularization parameter $ t \in \mathbb{R}_{\geq 0} $ is a delicate issue.Instead of solving the variational problem for a fixed parameter, the idea ofthe homotopy method is to compute a complete solution path $ u(t) $ as afunction of $ t $. In a celebrated paper by Osborne, Presnell, and Turlach, ithas been shown that the computational cost of this approach is often comparableto the cost of solving the corresponding least squares problem. Their analysisrelies on the one-at-a-time condition, which requires that different indicesenter or leave the support of the solution at distinct regularizationparameters. In this paper, we introduce a generalized homotopy algorithm basedon a nonnegative least squares problem, which does not require such acondition, and prove its termination after finitely many steps. At every pointof the path, we give a full characterization of all possible directions. Toillustrate our results, we discuss examples in which the standard homotopymethod either fails or becomes infeasible. To the best of our knowledge, ouralgorithm is the first to provably compute a full solution path for anarbitrary combination of an input matrix and a data vector.

The Homotopy Method Revisited: Computing Solution Paths of $\ell_1$-Regularized Problems