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

TL;DR

This paper introduces a generalized homotopy algorithm for computing complete solution paths of $\, ext{ extlbrack}1 extrbrack}$-regularized problems, overcoming limitations of previous methods and ensuring finite termination without the one-at-a-time condition.

Contribution

It presents the first provably finite homotopy algorithm for arbitrary matrices and data, removing the need for the one-at-a-time support change condition.

Findings

Algorithm terminates after finitely many steps.

Full characterization of solution directions at each path point.

Examples show standard methods can fail or be infeasible.

Abstract

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