On equivalence of ℓ1 norm based basic sparse representation problems

Jiang Rui; Qiao Hong; Zhang Bo

doi:10.1109/ICSPCC.2015.7338926

On equivalence of ℓ₁ norm based basic sparse representation problems

Jiang Rui
, Qiao Hong
, Zhang Bo

CAS - Institute of Automation
CAS - Institute of Applied Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

The ℓ₁ norm regularization problem, the ℓ₁ norm minimization problem and the ℓ₁ norm constraint problem are known collectively as the ℓ₁ norm based Basic Sparse Representation Problems (BSRPs), and have been popular basic models in the field of signal processing and machine learning. The equivalence of the above three problems is one of the crucial bases for the corresponding algorithms design. However, to the best our knowledge, this equivalence issue has not been addressed appropriately in the existing literature. In this paper, we will give a rigorous proof of the equivalence of the three ℓ₁ norm based BSRPs in the case when the dictionary is an overcomplete and row full rank matrix.

Original language	English
Title of host publication	2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015
Publisher	Institute of Electrical and Electronics Engineers Inc.
ISBN (Electronic)	9781479989188
DOIs	https://doi.org/10.1109/ICSPCC.2015.7338926
State	Published - 25 Nov 2015
Externally published	Yes
Event	5th IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015 - Ningbo, Zhejiang, China Duration: 19 Sep 2015 → 22 Sep 2015

Publication series

Name	2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015

Conference

Conference	5th IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015
Country/Territory	China
City	Ningbo, Zhejiang
Period	19/09/15 → 22/09/15

Keywords

Equivalence
ℓ norm constraint problem
ℓ norm minimization problem
ℓ norm regularization problem

Access to Document

10.1109/ICSPCC.2015.7338926

Cite this

Rui, J., Hong, Q., & Bo, Z. (2015). On equivalence of ℓ₁ norm based basic sparse representation problems. In 2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015 Article 7338926 (2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ICSPCC.2015.7338926

Rui, Jiang ; Hong, Qiao ; Bo, Zhang. / On equivalence of ℓ₁ norm based basic sparse representation problems. 2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015. Institute of Electrical and Electronics Engineers Inc., 2015. (2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015).

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abstract = "The ℓ1 norm regularization problem, the ℓ1 norm minimization problem and the ℓ1 norm constraint problem are known collectively as the ℓ1 norm based Basic Sparse Representation Problems (BSRPs), and have been popular basic models in the field of signal processing and machine learning. The equivalence of the above three problems is one of the crucial bases for the corresponding algorithms design. However, to the best our knowledge, this equivalence issue has not been addressed appropriately in the existing literature. In this paper, we will give a rigorous proof of the equivalence of the three ℓ1 norm based BSRPs in the case when the dictionary is an overcomplete and row full rank matrix.",

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Rui, J, Hong, Q & Bo, Z 2015, On equivalence of ℓ₁ norm based basic sparse representation problems. in 2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015., 7338926, 2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015, Institute of Electrical and Electronics Engineers Inc., 5th IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015, Ningbo, Zhejiang, China, 19/09/15. https://doi.org/10.1109/ICSPCC.2015.7338926

On equivalence of ℓ₁ norm based basic sparse representation problems. / Rui, Jiang; Hong, Qiao; Bo, Zhang.
2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015. Institute of Electrical and Electronics Engineers Inc., 2015. 7338926 (2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Rui J, Hong Q, Bo Z. On equivalence of ℓ₁ norm based basic sparse representation problems. In 2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015. Institute of Electrical and Electronics Engineers Inc. 2015. 7338926. (2015 IEEE International Conference on Signal Processing, Communications and Computing, ICSPCC 2015). doi: 10.1109/ICSPCC.2015.7338926