[{"date_created":"2019-07-22T13:23:48Z","doi":"10.1007/s10208-018-9395-y","date_published":"2019-06-01T00:00:00Z","page":"703-773","publication":"Foundations of Computational Mathematics","day":"01","year":"2019","oa":1,"publisher":"Springer","quality_controlled":"1","title":"Fundamental limits of weak recovery with applications to phase retrieval","external_id":{"arxiv":["1708.05932"]},"author":[{"first_name":"Marco","id":"27EB676C-8706-11E9-9510-7717E6697425","orcid":"0000-0002-3242-7020","full_name":"Mondelli, Marco","last_name":"Mondelli"},{"first_name":"Andrea","last_name":"Montanari","full_name":"Montanari, Andrea"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Mondelli, Marco, and Andrea Montanari. “Fundamental Limits of Weak Recovery with Applications to Phase Retrieval.” Foundations of Computational Mathematics, vol. 19, no. 3, Springer, 2019, pp. 703–73, doi:10.1007/s10208-018-9395-y.","ieee":"M. Mondelli and A. Montanari, “Fundamental limits of weak recovery with applications to phase retrieval,” Foundations of Computational Mathematics, vol. 19, no. 3. Springer, pp. 703–773, 2019.","short":"M. Mondelli, A. Montanari, Foundations of Computational Mathematics 19 (2019) 703–773.","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","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","chicago":"Mondelli, Marco, and Andrea Montanari. “Fundamental Limits of Weak Recovery with Applications to Phase Retrieval.” Foundations of Computational Mathematics. Springer, 2019. 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), 703–773."},"volume":19,"issue":"3","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"eissn":["1615-3383"]},"intvolume":" 19","month":"06","main_file_link":[{"url":"https://arxiv.org/abs/1708.05932","open_access":"1"}],"oa_version":"Preprint","abstract":[{"text":"In phase retrieval, we want to recover an unknown signal 𝑥∈ℂ𝑑 from n quadratic measurements of the form 𝑦𝑖=|⟨𝑎𝑖,𝑥⟩|2+𝑤𝑖, where 𝑎𝑖∈ℂ𝑑 are known sensing vectors and 𝑤𝑖 is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n needed to produce an estimator 𝑥^(𝑦) that is positively correlated with the signal 𝑥? We consider the case of Gaussian vectors 𝑎𝑎𝑖. We prove that—in the high-dimensional limit—a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For 𝑛≤𝑑−𝑜(𝑑), no estimator can do significantly better than random and achieve a strictly positive correlation. For 𝑛≥𝑑+𝑜(𝑑), 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 𝑦𝑖 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"}],"extern":"1","date_updated":"2021-01-12T08:08:28Z","status":"public","article_type":"original","type":"journal_article","_id":"6662"}]