The Condition Number in Phase Retrieval from Intensity Measurements
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911026067275776 |
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| author | Peng, Haiyang Han, Deren Huang, Meng |
| author_facet | Peng, Haiyang Han, Deren Huang, Meng |
| contents | This paper investigates the stability of phase retrieval by analyzing the condition number of the nonlinear map $Ψ_{\boldsymbol{A}}(\boldsymbol{x}) = \bigl(\lvert \langle {\boldsymbol{a}}_j, \boldsymbol{x} \rangle \rvert^2 \bigr)_{1 \le j \le m}$, where $\boldsymbol{a}_j \in \mathbb{H}^n$ are known sensing vectors with $\mathbb{H} \in \{\mathbb{R}, \mathbb{C}\}$. For each $p \ge 1$, we define the condition number $β_{Ψ_{\boldsymbol{A}}}^{\ell_p}$ as the ratio of optimal upper and lower Lipschitz constants of $Ψ_{\boldsymbol{A}}$ measured in the $\ell_p$ norm, with respect to the metric $\mathrm {dist}_\mathbb{H}\left(\boldsymbol{x}, \boldsymbol{y}\right) = \|\boldsymbol{x} \boldsymbol{x}^\ast - \boldsymbol{y} \boldsymbol{y}^\ast\|_*$. We establish universal lower bounds on $β_{Ψ_{\boldsymbol{A}}}^{\ell_p}$ for any sensing matrix $\boldsymbol{A} \in \mathbb{H}^{m \times d}$, proving that $β_{Ψ_{\boldsymbol{A}}}^{\ell_1} \ge π/2$ and $β_{Ψ_{\boldsymbol{A}}}^{\ell_2} \ge \sqrt{3}$ in the real case $(\mathbb{H} = \mathbb{R})$, and $β_{Ψ_{\boldsymbol{A}}}^{\ell_p} \ge 2$ for $p=1,2$ in the complex case $(\mathbb{H} = \mathbb{C})$. These bounds are shown to be asymptotically tight: both a deterministic harmonic frame $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ and Gaussian random matrices $\boldsymbol{A} \in \mathbb{H}^{m \times d}$ asymptotically attain them. Notably, the harmonic frame $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ achieves the optimal lower bound $\sqrt{3}$ for all $m \ge 3$ when $p=2$, thus serving as an optimal sensing matrix within $\boldsymbol{A} \in \mathbb{R}^{m \times 2}$. Our results provide the first explicit uniform lower bounds on $β_{Ψ_{\boldsymbol{A}}}^{\ell_p}$ and offer insights into the fundamental stability limits of phase retrieval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22053 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Condition Number in Phase Retrieval from Intensity Measurements Peng, Haiyang Han, Deren Huang, Meng Information Theory Functional Analysis 94A12, 65H10, 65F35 This paper investigates the stability of phase retrieval by analyzing the condition number of the nonlinear map $Ψ_{\boldsymbol{A}}(\boldsymbol{x}) = \bigl(\lvert \langle {\boldsymbol{a}}_j, \boldsymbol{x} \rangle \rvert^2 \bigr)_{1 \le j \le m}$, where $\boldsymbol{a}_j \in \mathbb{H}^n$ are known sensing vectors with $\mathbb{H} \in \{\mathbb{R}, \mathbb{C}\}$. For each $p \ge 1$, we define the condition number $β_{Ψ_{\boldsymbol{A}}}^{\ell_p}$ as the ratio of optimal upper and lower Lipschitz constants of $Ψ_{\boldsymbol{A}}$ measured in the $\ell_p$ norm, with respect to the metric $\mathrm {dist}_\mathbb{H}\left(\boldsymbol{x}, \boldsymbol{y}\right) = \|\boldsymbol{x} \boldsymbol{x}^\ast - \boldsymbol{y} \boldsymbol{y}^\ast\|_*$. We establish universal lower bounds on $β_{Ψ_{\boldsymbol{A}}}^{\ell_p}$ for any sensing matrix $\boldsymbol{A} \in \mathbb{H}^{m \times d}$, proving that $β_{Ψ_{\boldsymbol{A}}}^{\ell_1} \ge π/2$ and $β_{Ψ_{\boldsymbol{A}}}^{\ell_2} \ge \sqrt{3}$ in the real case $(\mathbb{H} = \mathbb{R})$, and $β_{Ψ_{\boldsymbol{A}}}^{\ell_p} \ge 2$ for $p=1,2$ in the complex case $(\mathbb{H} = \mathbb{C})$. These bounds are shown to be asymptotically tight: both a deterministic harmonic frame $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ and Gaussian random matrices $\boldsymbol{A} \in \mathbb{H}^{m \times d}$ asymptotically attain them. Notably, the harmonic frame $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ achieves the optimal lower bound $\sqrt{3}$ for all $m \ge 3$ when $p=2$, thus serving as an optimal sensing matrix within $\boldsymbol{A} \in \mathbb{R}^{m \times 2}$. Our results provide the first explicit uniform lower bounds on $β_{Ψ_{\boldsymbol{A}}}^{\ell_p}$ and offer insights into the fundamental stability limits of phase retrieval. |
| title | The Condition Number in Phase Retrieval from Intensity Measurements |
| topic | Information Theory Functional Analysis 94A12, 65H10, 65F35 |
| url | https://arxiv.org/abs/2506.22053 |