Complex Phase Retrieval via Conditional Quantiles: Finite-Sample Recovery without Noise Moments

Phase retrieval becomes statistically fragile when response noise is heavy-tailed or asymmetric: mean-based objectives can be dominated by extreme observations or even lose their population interpretation, while median-based methods fix attention on a single response quantile. We instead formulate complex phase retrieval through a prescribed conditional quantile, $Q_{\tau }(Y\mid a)=|a^{\ast }{x}_{\star }{|}^2,$ and propose robust quantile Wirtinger flow (RQWF). RQWF combines a clipped spectral estimate of the signal direction, quantile-based radial calibration, and bounded-score Wirtinger refinement, so that initialization and optimization target the same response functional. Under complex Gaussian sensing, we prove end-to-end finite-sample recovery assuming only local regularity of the noise distribution around the target quantile, with no noise-moment requirement. The resulting recovery rate is $\tilde{O}\left(\sqrt{\frac{n}{N}}\right)$, and a shifted-Cauchy submodel gives a matching dimension–sample lower bound up to logarithmic factors. The analysis also quantifies the bias introduced by smoothing and allows the same update sample to be reused throughout optimization. On a compute-matched Cauchy benchmark, RQWF-P lowers median recovery error relative to every tested complete SRPR/AdaIPL baseline in all 12 conditions and relative to Median-TWF in 10 of 12; prescribed non-median fits also improve selected asymmetric-noise cases. Student-$t$ gains depend on the comparator, while Gaussian and near-noiseless settings expose limits of the advantage. Together, the results show that conditional quantiles provide a tractable statistical target for complex phase retrieval precisely in regimes where moment-based formulations become unreliable.