强度测量下的相位检索中的条件数
信息论
2025-06-30 v1 泛函分析
math.IT
摘要
本文通过分析非线性映射 Ψ A ( x ) = ( ∣ ⟨ a j , x ⟩ ∣ 2 ) 1 ≤ j ≤ m \Psi_{\boldsymbol{A}}(\boldsymbol{x}) = \bigl(\lvert \langle {\boldsymbol{a}}_j, \boldsymbol{x} \rangle \rvert^2 \bigr)_{1 \le j \le m} Ψ A ( x ) = ( ∣⟨ a j , x ⟩ ∣ 2 ) 1 ≤ j ≤ m a j ∈ H n \boldsymbol{a}_j \in \mathbb{H}^n a j ∈ H n H ∈ { R , C } \mathbb{H} \in \{\mathbb{R}, \mathbb{C}\} H ∈ { R , C } p ≥ 1 p \ge 1 p ≥ 1 β Ψ A ℓ p \beta_{\Psi_{\boldsymbol{A}}}^{\ell_p} β Ψ A ℓ p Ψ A \Psi_{\boldsymbol{A}} Ψ A ℓ p \ell_p ℓ p d i s t H ( x , y ) = ∥ x x ∗ − y y ∗ ∥ ∗ \mathrm {dist}_\mathbb{H}\left(\boldsymbol{x}, \boldsymbol{y}\right) = \|\boldsymbol{x} \boldsymbol{x}^\ast - \boldsymbol{y} \boldsymbol{y}^\ast\|_* dist H ( x , y ) = ∥ x x ∗ − y y ∗ ∥ ∗ A ∈ H m × d \boldsymbol{A} \in \mathbb{H}^{m \times d} A ∈ H m × d β Ψ A ℓ p \beta_{\Psi_{\boldsymbol{A}}}^{\ell_p} β Ψ A ℓ p ( H = R ) (\mathbb{H} = \mathbb{R}) ( H = R ) β Ψ A ℓ 1 ≥ π / 2 \beta_{\Psi_{\boldsymbol{A}}}^{\ell_1} \ge \pi/2 β Ψ A ℓ 1 ≥ π /2 β Ψ A ℓ 2 ≥ 3 \beta_{\Psi_{\boldsymbol{A}}}^{\ell_2} \ge \sqrt{3} β Ψ A ℓ 2 ≥ 3 ( H = C ) (\mathbb{H} = \mathbb{C}) ( H = C ) p = 1 , 2 p=1,2 p = 1 , 2 β Ψ A ℓ p ≥ 2 \beta_{\Psi_{\boldsymbol{A}}}^{\ell_p} \ge 2 β Ψ A ℓ p ≥ 2 E m ∈ R m × 2 \boldsymbol{E}_m \in \mathbb{R}^{m \times 2} E m ∈ R m × 2 A ∈ H m × d \boldsymbol{A} \in \mathbb{H}^{m \times d} A ∈ H m × d E m ∈ R m × 2 \boldsymbol{E}_m \in \mathbb{R}^{m \times 2} E m ∈ R m × 2 p = 2 p=2 p = 2 m ≥ 3 m \ge 3 m ≥ 3 3 \sqrt{3} 3 A ∈ R m × 2 \boldsymbol{A} \in \mathbb{R}^{m \times 2} A ∈ R m × 2 β Ψ A ℓ p \beta_{\Psi_{\boldsymbol{A}}}^{\ell_p} β Ψ A ℓ p
引用
@article{arxiv.2506.22053,
title = {The Condition Number in Phase Retrieval from Intensity Measurements},
author = {Haiyang Peng and Deren Han and Meng Huang},
journal= {arXiv preprint arXiv:2506.22053},
year = {2025}
}