On the Hardness of the $L_1-L_2$ Regularization Problem
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911272220491776 |
|---|---|
| author | Ouyang, Yuyuan Yates, Kyle |
| author_facet | Ouyang, Yuyuan Yates, Kyle |
| contents | The sparse linear reconstruction problem is a core problem in signal processing which aims to recover sparse solutions to linear systems. The original problem regularized by the total number of nonzero components (also known as $L_0$ regularization) is well-known to be NP-hard. The relaxation of the $L_0$ regularization by using the $L_1$ norm offers a convex reformulation, but is only exact under certain conditions (e.g., restricted isometry property) which might be NP-hard to verify. To overcome the computational hardness of the $L_0$ regularization problem while providing tighter results than the $L_1$ relaxation, several alternate optimization problems have been proposed to find sparse solutions. One such problem is the $L_1-L_2$ minimization problem, which is to minimize the difference of the $L_1$ and $L_2$ norms subject to linear constraints. This paper proves that solving the $L_1-L_2$ minimization problem is NP-hard. Specifically, we prove that it is NP-hard to minimize the $L_1-L_2$ regularization function subject to linear constraints. Moreover, it is also NP-hard to solve the unconstrained formulation that minimizes the sum of a least squares term and the $L_1-L_2$ regularization function. Furthermore, restricting the feasible set to a smaller one by adding nonnegative constraints does not change the NP-hardness nature of the problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03216 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Hardness of the $L_1-L_2$ Regularization Problem Ouyang, Yuyuan Yates, Kyle Optimization and Control 68Q25, 94A12, 90C30 The sparse linear reconstruction problem is a core problem in signal processing which aims to recover sparse solutions to linear systems. The original problem regularized by the total number of nonzero components (also known as $L_0$ regularization) is well-known to be NP-hard. The relaxation of the $L_0$ regularization by using the $L_1$ norm offers a convex reformulation, but is only exact under certain conditions (e.g., restricted isometry property) which might be NP-hard to verify. To overcome the computational hardness of the $L_0$ regularization problem while providing tighter results than the $L_1$ relaxation, several alternate optimization problems have been proposed to find sparse solutions. One such problem is the $L_1-L_2$ minimization problem, which is to minimize the difference of the $L_1$ and $L_2$ norms subject to linear constraints. This paper proves that solving the $L_1-L_2$ minimization problem is NP-hard. Specifically, we prove that it is NP-hard to minimize the $L_1-L_2$ regularization function subject to linear constraints. Moreover, it is also NP-hard to solve the unconstrained formulation that minimizes the sum of a least squares term and the $L_1-L_2$ regularization function. Furthermore, restricting the feasible set to a smaller one by adding nonnegative constraints does not change the NP-hardness nature of the problems. |
| title | On the Hardness of the $L_1-L_2$ Regularization Problem |
| topic | Optimization and Control 68Q25, 94A12, 90C30 |
| url | https://arxiv.org/abs/2411.03216 |