论文标题
基于强度的模型的性能结合了嘈杂相位检索的模型
Performance bound of the intensity-based model for noisy phase retrieval
论文作者
论文摘要
嘈杂阶段检索的目的是估计信号$ \ MATHBF {X} _0 \ in \ MATHBB {C}^d $来自$ M $ M $ noisy强度测量$ B_J = \ left \ left \ left \ lvert \ langle \ langle \ langle \ mathbf {a} _j} _j} _j a} _j，\ mathbf} \ right \ rvert^2+η_j，\; j = 1，\ ldots，m $，其中$ \ mathbf {a} _j \ in \ mathbb {c}^d $是已知的测量向量，$η=（η_1，\ ldots，η_m）一个常用的模型，用于估计$ \ mathbf {x} _0 $是基于强度的模型$ \ wideHat {\ mathbf {x}}}：= \ \ mbox {argmin} _ {\ mathbf {x} \ langle \ mathbf {a} _j，\ mathbf {x} \ rangle \ right \ rvert \ rvert^2-b_j \ big）^2 $。尽管人们已经开发了许多有效的算法来解决基于强度的模型，但关于其估计性能的结果很少。在本文中，我们关注基于强度模型的估计性能，并证明该错误满足$ \ min_ {θ\ in \ Mathbb {r}}}} \ | \ | \ | \ wideHat {\ Mathbf {x}}}}}}}} - e^{iθ}}}}}}}}}}}}}} \ MATHBF {X}} {X} _ {x} _ _ \ | \ min \ big \ {\ frac {\ sqrt {\ |η\ | _2}}} {{m}^{1/4}}，\ frac {\ |η\ | | _2} {\ |在$ m \ gtrsim d $和$ \ gtrsim d $和$ \ mathbf {a} _j，j = 1，j = 1，\ ldots，m，m，m，m，$ be ldots，m，m，$ be be g gauss ver vectors的假设下，\ \ mathbf {x} _0 \ | _2 \ cdot \ cdot \ cdot \ sqrt \ sqrt {m}} \ big \} $。我们还表明，误差绑定是锋利的。对于$ \ mathbf {x} _0 $是$ s $ -sparse信号的情况，我们在假设$ M \ gtrsim s \ log（ed/s）$的情况下提出了类似的结果。据我们所知，我们的结果是基于强度的模型及其稀疏版本的第一个理论保证。我们的证明采用了门德尔森的小球方法，可以在非负经验过程中提供有效的下限。
The aim of noisy phase retrieval is to estimate a signal $\mathbf{x}_0\in \mathbb{C}^d$ from $m$ noisy intensity measurements $b_j=\left\lvert \langle \mathbf{a}_j,\mathbf{x}_0 \rangle \right\rvert^2+η_j, \; j=1,\ldots,m$, where $\mathbf{a}_j \in \mathbb{C}^d$ are known measurement vectors and $η=(η_1,\ldots,η_m)^\top \in \mathbb{R}^m$ is a noise vector. A commonly used model for estimating $\mathbf{x}_0$ is the intensity-based model $\widehat{\mathbf{x}}:=\mbox{argmin}_{\mathbf{x} \in \mathbb{C}^d} \sum_{j=1}^m \big(\left\lvert \langle \mathbf{a}_j,\mathbf{x} \rangle \right\rvert^2-b_j \big)^2$. Although one has already developed many efficient algorithms to solve the intensity-based model, there are very few results about its estimation performance. In this paper, we focus on the estimation performance of the intensity-based model and prove that the error bound satisfies $\min_{θ\in \mathbb{R}}\|\widehat{\mathbf{x}}-e^{iθ}\mathbf{x}_0\|_2 \lesssim \min\Big\{\frac{\sqrt{\|η\|_2}}{{m}^{1/4}}, \frac{\|η\|_2}{\| \mathbf{x}_0\|_2 \cdot \sqrt{m}}\Big\}$ under the assumption of $m \gtrsim d$ and $\mathbf{a}_j, j=1,\ldots,m,$ being Gaussian random vectors. We also show that the error bound is sharp. For the case where $\mathbf{x}_0$ is a $s$-sparse signal, we present a similar result under the assumption of $m \gtrsim s \log (ed/s)$. To the best of our knowledge, our results are the first theoretical guarantees for the intensity-based model and its sparse version. Our proofs employ Mendelson's small ball method which can deliver an effective lower bound on a nonnegative empirical process.
