The paper "PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming" by Emmanuel J. Candès, Thomas Strohmer, and Vladislav Voroninski addresses the problem of recovering a signal $\boldsymbol{x} \in \mathbb{C}^n$ from magnitude measurements of the form $|\langle \boldsymbol{x}, \boldsymbol{z}_i \rangle|^2$, where $\boldsymbol{z}_i$ are vectors sampled uniformly on the unit sphere. The authors prove that if the vectors $\boldsymbol{z}_i$ are independently and uniformly random, the signal $\boldsymbol{x}$ can be recovered exactly (up to a global phase factor) by solving a semidefinite program (SDP) or trace-norm minimization problem. This holds with high probability if the number of measurements $m$ is on the order of $n \log n$. The methodology is robust to additive noise, and the recovery error is small when the data are corrupted by noise. The paper also discusses the geometry of the problem and provides a detailed proof of the main theorem, including key lemmas and constructions of dual certificates.The paper "PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming" by Emmanuel J. Candès, Thomas Strohmer, and Vladislav Voroninski addresses the problem of recovering a signal $\boldsymbol{x} \in \mathbb{C}^n$ from magnitude measurements of the form $|\langle \boldsymbol{x}, \boldsymbol{z}_i \rangle|^2$, where $\boldsymbol{z}_i$ are vectors sampled uniformly on the unit sphere. The authors prove that if the vectors $\boldsymbol{z}_i$ are independently and uniformly random, the signal $\boldsymbol{x}$ can be recovered exactly (up to a global phase factor) by solving a semidefinite program (SDP) or trace-norm minimization problem. This holds with high probability if the number of measurements $m$ is on the order of $n \log n$. The methodology is robust to additive noise, and the recovery error is small when the data are corrupted by noise. The paper also discusses the geometry of the problem and provides a detailed proof of the main theorem, including key lemmas and constructions of dual certificates.