Bivariate retrieval from intensity of cross-correlation

Pulse characterization in ultra-fast optics presents a powerful motivation to study phase retrieval problems of high order. Frequency- and time-resolved techniques for pulse characterization both construct measurements that depend on the intensity of the cross-correlation between two unknown signals undergoing known modulations. The problem of recovering these signals has been traditionally studied and solved with alternating minimization, but recently Wirtinger gradient techniques were demonstrated to invert frequency-resolved measurements on a symmetric signal pair (Pinilla et al., 2019). In this paper, we construct a generalized Wirtinger gradient and Hessian to solve a wide breadth of problems including signal recovery from timeand frequency-resolved measurements. We further demonstrate that both measurement paradigms are special cases of low-rank phase retrieval but with a special structure that disrupts spectral initializers. To combat this problem, we present a tensor-based iterative hard thresholding initializer that, when paired with a Wirtinger gradient descent, is capable of recovering unknown signals with fewer measurements than matrix-based alternating minimization or spectral initialization methods. Finally, we employ Wirtinger gradient descent to recover signals from real-world DSCAN (Wilhelm et al., 2021) measurements and compare results with the existing state-of-the-art.