On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem

详细信息    查看全文

• 作者：Radu Balan^a ; ^{rvbalan@math.umd.edu" class="auth_mail" title="E-mail the corresponding author} ; Dongmian Zou^b ; ^{zou@math.umd.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A29 ; 65H10 ; 90C26
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：496
• 期：Complete
• 页码：152-181
• 全文大小：496 K

文摘

We prove two results with regard to reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”). First we show that phase retrievable nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Specifically, if nonlinear analysis maps $\alpha ,\beta :\hat{H}\to {\mathbb{R}}^m$ are injective, with $\alpha (x)={(|〈x,f_k〉|)}_{k=1}^m$ and $\beta (x)={(|〈x,f_k〉{|}^2)}_{k=1}^m$, where {f₁,…,f_m}$\{f_1,\dots ,f_m\}$ is a frame for a Hilbert space H   and $\hat{H}=H/T^1$, then α   is bi-Lipschitz with respect to the class of “natural metrics” D_p(x,y)=min_φ⁡‖x−e^iφy‖_p$D_p(x,y)={\min }_{\phi }{\Vert x-e^{i\phi }y\Vert }_p$, whereas β   is bi-Lipschitz with respect to the class of matrix-norm induced metrics d_p(x,y)=‖xx^⁎−yy^⁎‖_p$d_p(x,y)={\Vert x{x}^{⁎}-y{y}^{⁎}\Vert }_p$. Second we prove that reconstruction can be performed using Lipschitz continuous maps. That is, there exist left inverse maps (synthesis maps) $\omega ,\psi :{\mathbb{R}}^m\to \hat{H}$ of α and β respectively, that are Lipschitz continuous with respect to appropriate metrics. Additionally, we obtain the Lipschitz constants of ω and ψ in terms of the lower Lipschitz constants of α and β, respectively. Surprisingly, the increase in both Lipschitz constants is a relatively small factor, independent of the space dimension or the frame redundancy.