Computing the Proximal Operator of the $q$-th Power of the $\ell_{1,q}$-norm for Group Sparsity
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916773349031936 |
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| author | Lin, Rongrong Chen, Shihai Feng, Han Liu, Yulan |
| author_facet | Lin, Rongrong Chen, Shihai Feng, Han Liu, Yulan |
| contents | In this note, we comprehensively characterize the proximal operator of the $q$-th power of the $\ell_{1,q}$-norm (denoted by $\ell_{1,q}^{q}$) with $0\!<\!q\!<\!1$ by exploiting the well-known proximal operator of $|\cdot|^q$ on the real line. In particular, much more explicit characterizations can be obtained whenever $q\!=\!1/2$ and $q\!=\!2/3$ due to the existence of closed-form expressions for the proximal operators of $|\cdot|^{1/2}$ and $|\cdot|^{2/3}$. Numerical experiments demonstrate potential advantages of the $\ell_{1,q}^{q}$ regularization in the }inter-group and intra-group sparse vector recovery. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_14156 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing the Proximal Operator of the $q$-th Power of the $\ell_{1,q}$-norm for Group Sparsity Lin, Rongrong Chen, Shihai Feng, Han Liu, Yulan Numerical Analysis 65K10, 90C26 In this note, we comprehensively characterize the proximal operator of the $q$-th power of the $\ell_{1,q}$-norm (denoted by $\ell_{1,q}^{q}$) with $0\!<\!q\!<\!1$ by exploiting the well-known proximal operator of $|\cdot|^q$ on the real line. In particular, much more explicit characterizations can be obtained whenever $q\!=\!1/2$ and $q\!=\!2/3$ due to the existence of closed-form expressions for the proximal operators of $|\cdot|^{1/2}$ and $|\cdot|^{2/3}$. Numerical experiments demonstrate potential advantages of the $\ell_{1,q}^{q}$ regularization in the }inter-group and intra-group sparse vector recovery. |
| title | Computing the Proximal Operator of the $q$-th Power of the $\ell_{1,q}$-norm for Group Sparsity |
| topic | Numerical Analysis 65K10, 90C26 |
| url | https://arxiv.org/abs/2409.14156 |