{"intvolume":"        19","extern":"1","oa_version":"Preprint","oa":1,"publication_status":"published","article_type":"original","author":[{"last_name":"Mondelli","full_name":"Mondelli, Marco","first_name":"Marco","id":"27EB676C-8706-11E9-9510-7717E6697425","orcid":"0000-0002-3242-7020"},{"last_name":"Montanari","full_name":"Montanari, Andrea","first_name":"Andrea"}],"type":"journal_article","year":"2019","date_created":"2019-07-22T13:23:48Z","month":"06","date_updated":"2021-01-12T08:08:28Z","page":"703-773","volume":19,"_id":"6662","title":"Fundamental limits of weak recovery with applications to phase retrieval","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Springer","abstract":[{"lang":"eng","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."}],"issue":"3","language":[{"iso":"eng"}],"citation":{"ista":"Mondelli M, Montanari A. 2019. Fundamental limits of weak recovery with applications to phase retrieval. Foundations of Computational Mathematics. 19(3), 703–773.","mla":"Mondelli, Marco, and Andrea Montanari. “Fundamental Limits of Weak Recovery with Applications to Phase Retrieval.” <i>Foundations of Computational Mathematics</i>, vol. 19, no. 3, Springer, 2019, pp. 703–73, doi:<a href=\"https://doi.org/10.1007/s10208-018-9395-y\">10.1007/s10208-018-9395-y</a>.","short":"M. Mondelli, A. Montanari, Foundations of Computational Mathematics 19 (2019) 703–773.","apa":"Mondelli, M., &#38; Montanari, A. (2019). Fundamental limits of weak recovery with applications to phase retrieval. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-018-9395-y\">https://doi.org/10.1007/s10208-018-9395-y</a>","chicago":"Mondelli, Marco, and Andrea Montanari. “Fundamental Limits of Weak Recovery with Applications to Phase Retrieval.” <i>Foundations of Computational Mathematics</i>. Springer, 2019. <a href=\"https://doi.org/10.1007/s10208-018-9395-y\">https://doi.org/10.1007/s10208-018-9395-y</a>.","ama":"Mondelli M, Montanari A. Fundamental limits of weak recovery with applications to phase retrieval. <i>Foundations of Computational Mathematics</i>. 2019;19(3):703-773. doi:<a href=\"https://doi.org/10.1007/s10208-018-9395-y\">10.1007/s10208-018-9395-y</a>","ieee":"M. Mondelli and A. Montanari, “Fundamental limits of weak recovery with applications to phase retrieval,” <i>Foundations of Computational Mathematics</i>, vol. 19, no. 3. Springer, pp. 703–773, 2019."},"doi":"10.1007/s10208-018-9395-y","publication_identifier":{"eissn":["1615-3383"]},"date_published":"2019-06-01T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1708.05932"}],"day":"01","status":"public","publication":"Foundations of Computational Mathematics","quality_controlled":"1","external_id":{"arxiv":["1708.05932"]}}