Reconstruction with prior support information and non-Gaussian constraints
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929556544290816 |
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| author | Liu, Xiaotong Liang, Yiyu |
| author_facet | Liu, Xiaotong Liang, Yiyu |
| contents | In this study, we introduce a novel model, termed the Weighted Basis Pursuit Dequantization ($ω$-BPDQ$_p$), which incorporates prior support information by assigning weights on the $\ell_1$ norm in the $\ell_1$ minimization process and replaces the $\ell_2$ norm with the $\ell_p$ norm in the constraint. This adjustment addresses cases where noise deviates from a Gaussian distribution, such as quantized errors, which are common in practice. We demonstrate that Restricted Isometry Property (RIP$_{p,q}$) and Weighted Robust Null Space Property ($ω$-RNSP$_{p,q}$) ensure stable and robust reconstruction within $ω$-BPDQ$_p$, with the added observation that standard Gaussian random matrices satisfy these properties with high probability. Moreover, we establish a relationship between RIP$_{p,q}$ and $ω$-RNSP$_{p,q}$ that RIP$_{p,q}$ implies $ω$-RNSP$_{p,q}$. Additionally, numerical experiments confirm that the incorporation of weights and the non-Gaussian constraint results in improved reconstruction quality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reconstruction with prior support information and non-Gaussian constraints Liu, Xiaotong Liang, Yiyu Signal Processing Information Theory Classical Analysis and ODEs In this study, we introduce a novel model, termed the Weighted Basis Pursuit Dequantization ($ω$-BPDQ$_p$), which incorporates prior support information by assigning weights on the $\ell_1$ norm in the $\ell_1$ minimization process and replaces the $\ell_2$ norm with the $\ell_p$ norm in the constraint. This adjustment addresses cases where noise deviates from a Gaussian distribution, such as quantized errors, which are common in practice. We demonstrate that Restricted Isometry Property (RIP$_{p,q}$) and Weighted Robust Null Space Property ($ω$-RNSP$_{p,q}$) ensure stable and robust reconstruction within $ω$-BPDQ$_p$, with the added observation that standard Gaussian random matrices satisfy these properties with high probability. Moreover, we establish a relationship between RIP$_{p,q}$ and $ω$-RNSP$_{p,q}$ that RIP$_{p,q}$ implies $ω$-RNSP$_{p,q}$. Additionally, numerical experiments confirm that the incorporation of weights and the non-Gaussian constraint results in improved reconstruction quality. |
| title | Reconstruction with prior support information and non-Gaussian constraints |
| topic | Signal Processing Information Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2410.18116 |