The increasing dimensionality of data in the modern computing age presents new challenges and opportunities to the field of signal and data processing.  Here I will present some of the results we obtained through the use of heuristics statistical mechanics as well as rigorous methods for reconstructing a signal from a minimal number of generalized linear measurements, concentrating in particular to the phase retrieval problem. I shall discuss in particular the statistical and computational transitions, as well as the role of structured priors such as the one played by neutral networks.

Refs:
* Benjamin Aubin, Bruno Loureiro, Antoine Baker, Florent Krzakala, Lenka Zdeborová ; Proceedings of The First Mathematical and Scientific Machine Learning Conference, PMLR 107:55-73, 2020.
* Antoine Maillard, Bruno Loureiro, Florent Krzakala, Lenka Zdeborová, Phase retrieval in high dimensions: Statistical and computational phase transitions arXiv preprint arXiv:2006.05228