Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Lower Bound Convex Programs for Exact Sparse Optimization

Abstract : In exact sparse optimization problems on Rd (also known as sparsity constrained problems), one looks for solution that have few nonzero components. In this paper, we consider problems where sparsity is exactly measured either by the nonconvex l0 pseudonorm (and not by substitute penalty terms) or by the belonging of the solution to a finite union of subsets. Due to the combinatorial nature of the sparsity constraint, such problems do not generally display convexity properties, even if the criterion to minimize is convex. In the most common approach to tackle them, one replaces the sparsity constraint by a convex penalty term, supposed to induce sparsity. Thus doing, one loses the original exact sparse optimization problem, but gains convexity. However, by doing so, it is not clear that one obtains a lower bound of the original exact sparse optimization problem. In this paper, we propose another approach, where we lose convexity but where we gain at keeping the original exact sparse optimization formulation, by displaying lower bound convex minimization programs. For this purpose , we introduce suitable conjugacies, induced by a novel class of one-sided linear couplings. Thus equipped, we present a systematic way to design norms and lower bound convex minimization programs over their unit ball. The family of norms that we display encompasses most of the sparsity inducing norms used in machine learning. Therefore, our approach provides foundation and interpretation for their use.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

Cited literature [8 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02013990
Contributor : Michel de Lara <>
Submitted on : Monday, February 11, 2019 - 1:05:43 PM
Last modification on : Friday, July 5, 2019 - 9:28:02 AM
Long-term archiving on: : Sunday, May 12, 2019 - 2:06:48 PM

Files

HAL_sparse_optimization.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02013990, version 1
  • ARXIV : 1902.04813

Collections

Citation

Jean-Philippe Chancelier, Michel de Lara. Lower Bound Convex Programs for Exact Sparse Optimization. 2019. ⟨hal-02013990⟩

Share

Metrics

Record views

111

Files downloads

82