Stability in Phase Retrieval: Characterizing Condition Numbers and the Optimal Vector Set

Abstract

In this paper, we primarily focus on analyzing the stability property of phase retrieval by examining the bi-Lipschitz property of the map $\hat{I}{¦}_{\mathbf{A}}(\mathbf{x})=|\mathbf{A}\mathbf{x}|\in {\mathbb{R}}_{+}^m$, where $\mathbf{x}\in {\mathbb{H}}^d$ and $\mathbf{A}\in {\mathbb{H}}^{m\times d}$ is the measurement matrix for $\mathbb{H}\in \{\mathbb{R},\mathbb{C}\}$. We define the condition number ${\hat{I}}_{\mathbf{A}}^2=\frac{U_{\mathbf{A}}}{L_{\mathbf{A}}}$, where $L_{\mathbf{A}}$ and $U_{\mathbf{A}}$ represent the optimal lower and upper Lipschitz constants, respectively. We establish the first universal lower bound on ${\hat{I}}_{\mathbf{A}}^2$ by demonstrating that for any $\mathbf{A}\in {\mathbb{H}}^{m\times d}$, \begin{equation*} β_{\boldsymbol{A}}\geq β_0^{\mathbb{H}}=\begin{cases} \sqrt{\fracπ{π-2}}\,\,\approx\,\, 1.659 & \text{if $\mathbb{H}=\mathbb{R}$,}\\ \sqrt{\frac{4}{4-π}}\,\,\approx\,\, 2.159 & \text{if $\mathbb{H}=\mathbb{C}$.} \end{cases} \end{equation*} We prove that the condition number of a standard Gaussian matrix in ${\mathbb{H}}^{m\times d}$ asymptotically matches the lower bound $β_0^{\mathbb{H}}$ for both real and complex cases. This result indicates that the constant lower bound $β_0^{\mathbb{H}}$ is asymptotically tight, holding true for both the real and complex scenarios. As an application of this result, we utilize it to investigate the performance of quadratic models for phase retrieval. Lastly, we establish that for any odd integer $m\ge 3$, the harmonic frame $\mathbf{A}\in {\mathbb{R}}^{m\times 2}$ possesses the minimum condition number among all $\mathbf{A}\in {\mathbb{R}}^{m\times 2}$. We are confident that these findings carry substantial implications for enhancing our understanding of phase retrieval.