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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.04576 |
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| _version_ | 1866912414752047104 |
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| author | Huo, Haiye Xiao, Li |
| author_facet | Huo, Haiye Xiao, Li |
| contents | Sparse phase retrieval with redundant dictionary is to reconstruct the signals of interest that are (nearly) sparse in a redundant dictionary or frame from the phaseless measurements via the optimization models. Gao [7] presented conditions on the measurement matrix, called null space property (NSP) and strong dictionary restricted isometry property (S-DRIP), for exact and stable recovery of dictionary-$k$-sparse signals via the $\ell_1$-analysis model for sparse phase retrieval with redundant dictionary, respectively, where, in particularly, the S-DRIP of order $tk$ with $t>1$ was derived. In this paper, motivated by many advantages of the $\ell_q$ minimization with $0<q\leq1$, e.g., reduction of the number of measurements required, we generalize these two conditions to the $\ell_q$-analysis model. Specifically, we first present two NSP variants for exact recovery of dictionary-$k$-sparse signals via the $\ell_q$-analysis model in the noiseless scenario. Moreover, we investigate the S-DRIP of order $tk$ with $0<t<\frac{4}{3}$ for stable recovery of dictionary-$k$-sparse signals via the $\ell_q$-analysis model in the noisy scenario, which will complement the existing result of the S-DRIP of order $tk$ with $t\geq2$ obtained in [4]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04576 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparse Phase Retrieval with Redundant Dictionary via $\ell_q (0<q\le 1)$-Analysis Model Huo, Haiye Xiao, Li Information Theory Sparse phase retrieval with redundant dictionary is to reconstruct the signals of interest that are (nearly) sparse in a redundant dictionary or frame from the phaseless measurements via the optimization models. Gao [7] presented conditions on the measurement matrix, called null space property (NSP) and strong dictionary restricted isometry property (S-DRIP), for exact and stable recovery of dictionary-$k$-sparse signals via the $\ell_1$-analysis model for sparse phase retrieval with redundant dictionary, respectively, where, in particularly, the S-DRIP of order $tk$ with $t>1$ was derived. In this paper, motivated by many advantages of the $\ell_q$ minimization with $0<q\leq1$, e.g., reduction of the number of measurements required, we generalize these two conditions to the $\ell_q$-analysis model. Specifically, we first present two NSP variants for exact recovery of dictionary-$k$-sparse signals via the $\ell_q$-analysis model in the noiseless scenario. Moreover, we investigate the S-DRIP of order $tk$ with $0<t<\frac{4}{3}$ for stable recovery of dictionary-$k$-sparse signals via the $\ell_q$-analysis model in the noisy scenario, which will complement the existing result of the S-DRIP of order $tk$ with $t\geq2$ obtained in [4]. |
| title | Sparse Phase Retrieval with Redundant Dictionary via $\ell_q (0<q\le 1)$-Analysis Model |
| topic | Information Theory |
| url | https://arxiv.org/abs/2506.04576 |