Phase retrieval with rank $d$ measurements -- \emph{descending} algorithms phase transitions
This work provides theoretical insights into phase retrieval algorithms for signal processing and imaging applications, but it is incremental as it builds on prior RDT-based analysis.
The paper tackles the problem of phase retrieval with rank d measurements by generalizing a Random Duality Theory (RDT) framework to analyze descending algorithms, showing that the minimal sample complexity ratio for success exhibits a phase transition phenomenon, with simulated results in small dimensions (e.g., problem sizes around 100) closely matching theoretical predictions.
Companion paper [118] developed a powerful \emph{Random duality theory} (RDT) based analytical program to statistically characterize performance of \emph{descending} phase retrieval algorithms (dPR) (these include all variants of gradient descents and among them widely popular Wirtinger flows). We here generalize the program and show how it can be utilized to handle rank $d$ positive definite phase retrieval (PR) measurements (with special cases $d=1$ and $d=2$ serving as emulations of the real and complex phase retrievals, respectively). In particular, we observe that the minimal sample complexity ratio (number of measurements scaled by the dimension of the unknown signal) which ensures dPR's success exhibits a phase transition (PT) phenomenon. For both plain and lifted RDT we determine phase transitions locations. To complement theoretical results we implement a log barrier gradient descent variant and observe that, even in small dimensional scenarios (with problem sizes on the order of 100), the simulated phase transitions are in an excellent agreement with the theoretical predictions.