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The homotopy method revisited: Computing solution paths of $ \ell_1$-regularized problems

Authors: Bjoern Bringmann, Daniel Cremers, Felix Krahmer and Michael Moeller
Journal: Math. Comp. 87 (2018), 2343-2364
MSC (2010): Primary 49N45, 90C25, 62J07
Published electronically: December 14, 2017
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Abstract: $ \ell _1 $-regularized linear inverse problems frequently arise 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 their celebrated paper A new approach to variable selection in least squares problems [IMA J. Numer. Anal. 20 (2000), no. 3, 389-403], Osborne, Presnell, and Turlach showed 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 piecewise linear and continuous solution path for an arbitrary combination of a measurement matrix and a data vector.

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Additional Information

Bjoern Bringmann
Affiliation: Department of Mathematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles, California 90095

Daniel Cremers
Affiliation: Department of Computer Science, Technische Universität München, Informatik 9, Boltzmannstr. 3, D-85748 Garching bei München

Felix Krahmer
Affiliation: Department of Mathematics, Technische Universität München, Boltzmannstr. 3, D-85748 Garching bei München

Michael Moeller
Affiliation: Department of Electrical Engineering and Computer Science, Universität Siegen, Hölderlinstr. 3, D-57076 Siegen

Keywords: Lasso, $\ell_1$-regularization, $\ell_1$-norm, homotopy, compressed sensing, nonnegative least squares, convex optimization
Received by editor(s): May 30, 2016
Received by editor(s) in revised form: April 6, 2017
Published electronically: December 14, 2017
Additional Notes: The second and fourth authors were supported by the ERC Starting Grant “ConvexVision”.
The third author’s contribution was supported by the German Science Foundation DFG in context of the Emmy Noether junior research group KR 4512/1-1 (RaSenQuaSI)
Article copyright: © Copyright 2017 American Mathematical Society

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