{"intvolume":" 19","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1708.05932"}],"quality_controlled":"1","title":"Fundamental limits of weak recovery with applications to phase retrieval","issue":"3","oa_version":"Preprint","article_type":"original","year":"2019","publisher":"Springer","day":"01","date_published":"2019-06-01T00:00:00Z","date_updated":"2021-01-12T08:08:28Z","volume":19,"extern":"1","type":"journal_article","publication_identifier":{"eissn":["1615-3383"]},"_id":"6662","author":[{"full_name":"Mondelli, Marco","first_name":"Marco","orcid":"0000-0002-3242-7020","last_name":"Mondelli","id":"27EB676C-8706-11E9-9510-7717E6697425"},{"first_name":"Andrea","full_name":"Montanari, Andrea","last_name":"Montanari"}],"doi":"10.1007/s10208-018-9395-y","publication_status":"published","publication":"Foundations of Computational Mathematics","abstract":[{"text":"In phase retrieval, we want to recover an unknown signal π₯ββπ from n quadratic measurements of the form π¦π=|β¨ππ,π₯β©|2+π€π, where ππββπ are known sensing vectors and π€π is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n needed to produce an estimator π₯^(π¦) that is positively correlated with the signal π₯? We consider the case of Gaussian vectors πππ. We prove thatβin the high-dimensional limitβa sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For πβ€πβπ(π), no estimator can do significantly better than random and achieve a strictly positive correlation. For πβ₯π+π(π), a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on classical information-theoretic arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper bound and lower bound generalize beyond phase retrieval to measurements π¦π produced according to a generalized linear model. As a by-product of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.","lang":"eng"}],"citation":{"apa":"Mondelli, M., & Montanari, A. (2019). Fundamental limits of weak recovery with applications to phase retrieval. Foundations of Computational Mathematics. Springer. https://doi.org/10.1007/s10208-018-9395-y","ista":"Mondelli M, Montanari A. 2019. Fundamental limits of weak recovery with applications to phase retrieval. Foundations of Computational Mathematics. 19(3), 703β773.","chicago":"Mondelli, Marco, and Andrea Montanari. βFundamental Limits of Weak Recovery with Applications to Phase Retrieval.β Foundations of Computational Mathematics. Springer, 2019. https://doi.org/10.1007/s10208-018-9395-y.","ieee":"M. Mondelli and A. Montanari, βFundamental limits of weak recovery with applications to phase retrieval,β Foundations of Computational Mathematics, vol. 19, no. 3. Springer, pp. 703β773, 2019.","ama":"Mondelli M, Montanari A. Fundamental limits of weak recovery with applications to phase retrieval. Foundations of Computational Mathematics. 2019;19(3):703-773. doi:10.1007/s10208-018-9395-y","mla":"Mondelli, Marco, and Andrea Montanari. βFundamental Limits of Weak Recovery with Applications to Phase Retrieval.β Foundations of Computational Mathematics, vol. 19, no. 3, Springer, 2019, pp. 703β73, doi:10.1007/s10208-018-9395-y.","short":"M. Mondelli, A. Montanari, Foundations of Computational Mathematics 19 (2019) 703β773."},"month":"06","date_created":"2019-07-22T13:23:48Z","status":"public","page":"703-773","external_id":{"arxiv":["1708.05932"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1}