[{"volume":33,"acknowledgement":"The first author is partially supported by NSF Grant DMS-2113489 and grateful for the AMS-SIMONS travel grant (2020–2023). The second author is supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.\r\nThe authors would like to thank the Editor, Associate Editor and an anonymous referee for their many critical suggestions which have significantly improved the paper. We also want to thank Zhigang Bao and Ji Oon Lee for many helpful discussions and comments.","arxiv":1,"doi":"10.1214/22-aap1882","day":"01","abstract":[{"lang":"eng","text":"Consider the random matrix model A1/2UBU∗A1/2, where A and B are two N × N deterministic matrices and U is either an N × N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201–220), the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution\r\nof the limiting ESDs of A and B, denoted as μα \u0002 μβ, where μα and μβ are the limiting ESDs of A and B, respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of μA \u0002μB, where μA and μB are the ESDs of A and B, respectively and the associated subordination functions\r\nhave a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of A, B and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the additive model A+UBU∗ in (J. Funct. Anal. 271 (2016) 672–719; Comm. Math.\r\nPhys. 349 (2017) 947–990; Adv. Math. 319 (2017) 251–291; J. Funct. Anal. 279 (2020) 108639), and our results can be regarded as the counterparts of (J. Funct. Anal. 279 (2020) 108639) for the multiplicative model. "}],"date_updated":"2024-01-09T08:16:41Z","citation":{"ieee":"X. Ding and H. C. Ji, “Local laws for multiplication of random matrices,” <i>The Annals of Applied Probability</i>, vol. 33, no. 4. Institute of Mathematical Statistics, pp. 2981–3009, 2023.","chicago":"Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.” <i>The Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2023. <a href=\"https://doi.org/10.1214/22-aap1882\">https://doi.org/10.1214/22-aap1882</a>.","apa":"Ding, X., &#38; Ji, H. C. (2023). Local laws for multiplication of random matrices. <i>The Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/22-aap1882\">https://doi.org/10.1214/22-aap1882</a>","ama":"Ding X, Ji HC. Local laws for multiplication of random matrices. <i>The Annals of Applied Probability</i>. 2023;33(4):2981-3009. doi:<a href=\"https://doi.org/10.1214/22-aap1882\">10.1214/22-aap1882</a>","ista":"Ding X, Ji HC. 2023. Local laws for multiplication of random matrices. The Annals of Applied Probability. 33(4), 2981–3009.","mla":"Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.” <i>The Annals of Applied Probability</i>, vol. 33, no. 4, Institute of Mathematical Statistics, 2023, pp. 2981–3009, doi:<a href=\"https://doi.org/10.1214/22-aap1882\">10.1214/22-aap1882</a>.","short":"X. Ding, H.C. Ji, The Annals of Applied Probability 33 (2023) 2981–3009."},"year":"2023","external_id":{"arxiv":["2010.16083"]},"publisher":"Institute of Mathematical Statistics","article_type":"original","page":"2981-3009","quality_controlled":"1","ec_funded":1,"publication_status":"published","date_created":"2024-01-08T13:03:18Z","department":[{"_id":"LaEr"}],"article_processing_charge":"No","title":"Local laws for multiplication of random matrices","intvolume":"        33","_id":"14750","scopus_import":"1","author":[{"last_name":"Ding","first_name":"Xiucai","full_name":"Ding, Xiucai"},{"full_name":"Ji, Hong Chang","first_name":"Hong Chang","last_name":"Ji","id":"dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d"}],"issue":"4","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2010.16083","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","publication_identifier":{"issn":["1050-5164"]},"oa":1,"date_published":"2023-08-01T00:00:00Z","type":"journal_article","language":[{"iso":"eng"}],"keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"oa_version":"Preprint","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"month":"08","publication":"The Annals of Applied Probability"},{"day":"01","doi":"10.1214/22-aap1826","arxiv":1,"abstract":[{"text":"We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.","lang":"eng"}],"citation":{"ama":"Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. <i>The Annals of Applied Probability</i>. 2023;33(1):677-725. doi:<a href=\"https://doi.org/10.1214/22-aap1826\">10.1214/22-aap1826</a>","apa":"Schnelli, K., &#38; Xu, Y. (2023). Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. <i>The Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/22-aap1826\">https://doi.org/10.1214/22-aap1826</a>","ieee":"K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices,” <i>The Annals of Applied Probability</i>, vol. 33, no. 1. Institute of Mathematical Statistics, pp. 677–725, 2023.","chicago":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” <i>The Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2023. <a href=\"https://doi.org/10.1214/22-aap1826\">https://doi.org/10.1214/22-aap1826</a>.","short":"K. Schnelli, Y. Xu, The Annals of Applied Probability 33 (2023) 677–725.","mla":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” <i>The Annals of Applied Probability</i>, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp. 677–725, doi:<a href=\"https://doi.org/10.1214/22-aap1826\">10.1214/22-aap1826</a>.","ista":"Schnelli K, Xu Y. 2023. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. 33(1), 677–725."},"year":"2023","date_updated":"2024-01-10T13:31:46Z","external_id":{"arxiv":["2108.02728"],"isi":["000946432400021"]},"isi":1,"volume":33,"acknowledgement":"K. Schnelli was supported by the Swedish Research Council Grants VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Y. Xu was supported by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond” No. 101020331.","department":[{"_id":"LaEr"}],"date_created":"2024-01-10T09:23:31Z","article_processing_charge":"No","publication_status":"published","intvolume":"        33","title":"Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices","scopus_import":"1","_id":"14775","issue":"1","author":[{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","last_name":"Schnelli","first_name":"Kevin"},{"last_name":"Xu","first_name":"Yuanyuan","full_name":"Xu, Yuanyuan","orcid":"0000-0003-1559-1205","id":"7902bdb1-a2a4-11eb-a164-c9216f71aea3"}],"publisher":"Institute of Mathematical Statistics","article_type":"original","quality_controlled":"1","ec_funded":1,"page":"677-725","publication_identifier":{"issn":["1050-5164"]},"oa":1,"type":"journal_article","date_published":"2023-02-01T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2108.02728","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d","call_identifier":"H2020"}],"oa_version":"Preprint","month":"02","publication":"The Annals of Applied Probability","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"language":[{"iso":"eng"}]},{"publication_identifier":{"issn":["0091-1798"]},"oa":1,"type":"journal_article","date_published":"2023-11-01T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2206.04448"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d","call_identifier":"H2020"}],"oa_version":"Preprint","month":"11","publication":"The Annals of Probability","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"language":[{"iso":"eng"}],"day":"01","arxiv":1,"doi":"10.1214/23-aop1643","abstract":[{"text":"We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an n×n random matrix with independent identically distributed complex entries as n tends to infinity. All terms in the expansion are universal.","lang":"eng"}],"year":"2023","citation":{"short":"G. Cipolloni, L. Erdös, D.J. Schröder, Y. Xu, The Annals of Probability 51 (2023) 2192–2242.","mla":"Cipolloni, Giorgio, et al. “On the Rightmost Eigenvalue of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>, vol. 51, no. 6, Institute of Mathematical Statistics, 2023, pp. 2192–242, doi:<a href=\"https://doi.org/10.1214/23-aop1643\">10.1214/23-aop1643</a>.","ista":"Cipolloni G, Erdös L, Schröder DJ, Xu Y. 2023. On the rightmost eigenvalue of non-Hermitian random matrices. The Annals of Probability. 51(6), 2192–2242.","ama":"Cipolloni G, Erdös L, Schröder DJ, Xu Y. On the rightmost eigenvalue of non-Hermitian random matrices. <i>The Annals of Probability</i>. 2023;51(6):2192-2242. doi:<a href=\"https://doi.org/10.1214/23-aop1643\">10.1214/23-aop1643</a>","apa":"Cipolloni, G., Erdös, L., Schröder, D. J., &#38; Xu, Y. (2023). On the rightmost eigenvalue of non-Hermitian random matrices. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/23-aop1643\">https://doi.org/10.1214/23-aop1643</a>","ieee":"G. Cipolloni, L. Erdös, D. J. Schröder, and Y. Xu, “On the rightmost eigenvalue of non-Hermitian random matrices,” <i>The Annals of Probability</i>, vol. 51, no. 6. Institute of Mathematical Statistics, pp. 2192–2242, 2023.","chicago":"Cipolloni, Giorgio, László Erdös, Dominik J Schröder, and Yuanyuan Xu. “On the Rightmost Eigenvalue of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2023. <a href=\"https://doi.org/10.1214/23-aop1643\">https://doi.org/10.1214/23-aop1643</a>."},"date_updated":"2024-01-23T10:56:30Z","external_id":{"arxiv":["2206.04448"]},"acknowledgement":"The second and the fourth author were supported by the ERC Advanced Grant\r\n“RMTBeyond” No. 101020331. The third author was supported by Dr. Max Rössler, the\r\nWalter Haefner Foundation and the ETH Zürich Foundation.","volume":51,"article_processing_charge":"No","date_created":"2024-01-22T08:08:41Z","department":[{"_id":"LaEr"}],"publication_status":"published","intvolume":"        51","title":"On the rightmost eigenvalue of non-Hermitian random matrices","_id":"14849","issue":"6","author":[{"orcid":"0000-0002-4901-7992","full_name":"Cipolloni, Giorgio","first_name":"Giorgio","last_name":"Cipolloni","id":"42198EFA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"id":"408ED176-F248-11E8-B48F-1D18A9856A87","last_name":"Schröder","first_name":"Dominik J","full_name":"Schröder, Dominik J","orcid":"0000-0002-2904-1856"},{"full_name":"Xu, Yuanyuan","first_name":"Yuanyuan","last_name":"Xu"}],"publisher":"Institute of Mathematical Statistics","article_type":"original","quality_controlled":"1","ec_funded":1,"page":"2192-2242"},{"volume":11,"date_updated":"2023-08-03T06:32:22Z","year":"2022","citation":{"apa":"Reker, J. (2022). On the operator norm of a Hermitian random matrix with correlated entries. <i>Random Matrices: Theory and Applications</i>. World Scientific. <a href=\"https://doi.org/10.1142/s2010326322500368\">https://doi.org/10.1142/s2010326322500368</a>","ama":"Reker J. On the operator norm of a Hermitian random matrix with correlated entries. <i>Random Matrices: Theory and Applications</i>. 2022;11(4). doi:<a href=\"https://doi.org/10.1142/s2010326322500368\">10.1142/s2010326322500368</a>","ieee":"J. Reker, “On the operator norm of a Hermitian random matrix with correlated entries,” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4. World Scientific, 2022.","chicago":"Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated Entries.” <i>Random Matrices: Theory and Applications</i>. World Scientific, 2022. <a href=\"https://doi.org/10.1142/s2010326322500368\">https://doi.org/10.1142/s2010326322500368</a>.","short":"J. Reker, Random Matrices: Theory and Applications 11 (2022).","mla":"Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated Entries.” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4, 2250036, World Scientific, 2022, doi:<a href=\"https://doi.org/10.1142/s2010326322500368\">10.1142/s2010326322500368</a>.","ista":"Reker J. 2022. On the operator norm of a Hermitian random matrix with correlated entries. Random Matrices: Theory and Applications. 11(4), 2250036."},"isi":1,"external_id":{"arxiv":["2103.03906"],"isi":["000848873800001"]},"doi":"10.1142/s2010326322500368","arxiv":1,"day":"01","abstract":[{"lang":"eng","text":"We consider a correlated NxN Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one."}],"quality_controlled":"1","publisher":"World Scientific","article_type":"original","_id":"11135","scopus_import":"1","author":[{"id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","full_name":"Reker, Jana","last_name":"Reker","first_name":"Jana"}],"issue":"4","publication_status":"published","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"article_processing_charge":"No","date_created":"2022-04-08T07:11:12Z","title":"On the operator norm of a Hermitian random matrix with correlated entries","intvolume":"        11","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2103.03906","open_access":"1"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","date_published":"2022-10-01T00:00:00Z","type":"journal_article","publication_identifier":{"eissn":["2010-3271"],"issn":["2010-3263"]},"oa":1,"language":[{"iso":"eng"}],"keyword":["Discrete Mathematics and Combinatorics","Statistics","Probability and Uncertainty","Statistics and Probability","Algebra and Number Theory"],"publication":"Random Matrices: Theory and Applications","oa_version":"Preprint","month":"10","article_number":"2250036"},{"publication":"Electronic Journal of Probability","has_accepted_license":"1","month":"09","oa_version":"Published Version","project":[{"call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331"}],"language":[{"iso":"eng"}],"keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"date_published":"2022-09-12T00:00:00Z","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"oa":1,"publication_identifier":{"eissn":["1083-6489"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","file":[{"success":1,"access_level":"open_access","relation":"main_file","creator":"dernst","file_id":"12464","checksum":"bb647b48fbdb59361210e425c220cdcb","file_size":502149,"date_created":"2023-01-30T11:59:21Z","content_type":"application/pdf","file_name":"2022_ElecJournProbability_Cipolloni.pdf","date_updated":"2023-01-30T11:59:21Z"}],"author":[{"first_name":"Giorgio","last_name":"Cipolloni","orcid":"0000-0002-4901-7992","full_name":"Cipolloni, Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"id":"408ED176-F248-11E8-B48F-1D18A9856A87","last_name":"Schröder","first_name":"Dominik J","full_name":"Schröder, Dominik J","orcid":"0000-0002-2904-1856"}],"_id":"12290","scopus_import":"1","title":"Optimal multi-resolvent local laws for Wigner matrices","intvolume":"        27","publication_status":"published","department":[{"_id":"LaEr"}],"date_created":"2023-01-16T10:04:38Z","article_processing_charge":"No","file_date_updated":"2023-01-30T11:59:21Z","page":"1-38","quality_controlled":"1","ec_funded":1,"article_type":"original","publisher":"Institute of Mathematical Statistics","isi":1,"external_id":{"isi":["000910863700003"]},"date_updated":"2023-08-04T10:32:23Z","citation":{"ista":"Cipolloni G, Erdös L, Schröder DJ. 2022. Optimal multi-resolvent local laws for Wigner matrices. Electronic Journal of Probability. 27, 1–38.","mla":"Cipolloni, Giorgio, et al. “Optimal Multi-Resolvent Local Laws for Wigner Matrices.” <i>Electronic Journal of Probability</i>, vol. 27, Institute of Mathematical Statistics, 2022, pp. 1–38, doi:<a href=\"https://doi.org/10.1214/22-ejp838\">10.1214/22-ejp838</a>.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Electronic Journal of Probability 27 (2022) 1–38.","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Optimal multi-resolvent local laws for Wigner matrices,” <i>Electronic Journal of Probability</i>, vol. 27. Institute of Mathematical Statistics, pp. 1–38, 2022.","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Optimal Multi-Resolvent Local Laws for Wigner Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2022. <a href=\"https://doi.org/10.1214/22-ejp838\">https://doi.org/10.1214/22-ejp838</a>.","ama":"Cipolloni G, Erdös L, Schröder DJ. Optimal multi-resolvent local laws for Wigner matrices. <i>Electronic Journal of Probability</i>. 2022;27:1-38. doi:<a href=\"https://doi.org/10.1214/22-ejp838\">10.1214/22-ejp838</a>","apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2022). Optimal multi-resolvent local laws for Wigner matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/22-ejp838\">https://doi.org/10.1214/22-ejp838</a>"},"year":"2022","abstract":[{"lang":"eng","text":"We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale."}],"doi":"10.1214/22-ejp838","day":"12","ddc":["510"],"volume":27,"acknowledgement":"L. Erdős was supported by ERC Advanced Grant “RMTBeyond” No. 101020331. D. Schröder was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation."},{"date_published":"2022-11-24T00:00:00Z","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"oa":1,"publication_identifier":{"issn":["1742-5468"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","related_material":{"record":[{"status":"public","id":"10598","relation":"earlier_version"}]},"status":"public","file":[{"relation":"main_file","access_level":"open_access","success":1,"file_id":"12481","creator":"dernst","date_created":"2023-02-02T08:35:52Z","file_size":1729997,"checksum":"01411ffa76d3e380a0446baeb89b1ef7","date_updated":"2023-02-02T08:35:52Z","file_name":"2022_JourStatisticalMechanics_Mondelli.pdf","content_type":"application/pdf"}],"publication":"Journal of Statistical Mechanics: Theory and Experiment","has_accepted_license":"1","month":"11","article_number":"114003","oa_version":"Published Version","project":[{"_id":"059876FA-7A3F-11EA-A408-12923DDC885E","name":"Prix Lopez-Loretta 2019 - Marco Mondelli"}],"language":[{"iso":"eng"}],"keyword":["Statistics","Probability and Uncertainty","Statistics and Probability","Statistical and Nonlinear Physics"],"isi":1,"external_id":{"isi":["000889589900001"]},"date_updated":"2024-03-07T10:36:52Z","citation":{"apa":"Mondelli, M., &#38; Venkataramanan, R. (2022). Approximate message passing with spectral initialization for generalized linear models. <i>Journal of Statistical Mechanics: Theory and Experiment</i>. IOP Publishing. <a href=\"https://doi.org/10.1088/1742-5468/ac9828\">https://doi.org/10.1088/1742-5468/ac9828</a>","ama":"Mondelli M, Venkataramanan R. Approximate message passing with spectral initialization for generalized linear models. <i>Journal of Statistical Mechanics: Theory and Experiment</i>. 2022;2022(11). doi:<a href=\"https://doi.org/10.1088/1742-5468/ac9828\">10.1088/1742-5468/ac9828</a>","chicago":"Mondelli, Marco, and Ramji Venkataramanan. “Approximate Message Passing with Spectral Initialization for Generalized Linear Models.” <i>Journal of Statistical Mechanics: Theory and Experiment</i>. IOP Publishing, 2022. <a href=\"https://doi.org/10.1088/1742-5468/ac9828\">https://doi.org/10.1088/1742-5468/ac9828</a>.","ieee":"M. Mondelli and R. Venkataramanan, “Approximate message passing with spectral initialization for generalized linear models,” <i>Journal of Statistical Mechanics: Theory and Experiment</i>, vol. 2022, no. 11. IOP Publishing, 2022.","mla":"Mondelli, Marco, and Ramji Venkataramanan. “Approximate Message Passing with Spectral Initialization for Generalized Linear Models.” <i>Journal of Statistical Mechanics: Theory and Experiment</i>, vol. 2022, no. 11, 114003, IOP Publishing, 2022, doi:<a href=\"https://doi.org/10.1088/1742-5468/ac9828\">10.1088/1742-5468/ac9828</a>.","short":"M. Mondelli, R. Venkataramanan, Journal of Statistical Mechanics: Theory and Experiment 2022 (2022).","ista":"Mondelli M, Venkataramanan R. 2022. Approximate message passing with spectral initialization for generalized linear models. Journal of Statistical Mechanics: Theory and Experiment. 2022(11), 114003."},"year":"2022","abstract":[{"lang":"eng","text":"We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms with many appealing features: the performance of AMP in the high-dimensional limit can be succinctly characterized under suitable model assumptions; AMP can also be tailored to the empirical distribution of the signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms. However, a major issue of AMP is that in many models (such as phase retrieval), it requires an initialization correlated with the ground-truth signal and independent from the measurement matrix. Assuming that such an initialization is available is typically not realistic. In this paper, we solve this problem by proposing an AMP algorithm initialized with a spectral estimator. With such an initialization, the standard AMP analysis fails since the spectral estimator depends in a complicated way on the design matrix. Our main contribution is a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. The key technical idea is to define and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP. We also provide numerical results that demonstrate the validity of the proposed approach."}],"doi":"10.1088/1742-5468/ac9828","day":"24","ddc":["510","530"],"acknowledgement":"The authors would like to thank Andrea Montanari for helpful discussions.\r\nM Mondelli was partially supported by the 2019 Lopez-Loreta Prize. R Venkataramanan was partially supported by the Alan Turing Institute under the EPSRC Grant\r\nEP/N510129/1.","volume":2022,"author":[{"id":"27EB676C-8706-11E9-9510-7717E6697425","orcid":"0000-0002-3242-7020","full_name":"Mondelli, Marco","first_name":"Marco","last_name":"Mondelli"},{"full_name":"Venkataramanan, Ramji","last_name":"Venkataramanan","first_name":"Ramji"}],"issue":"11","_id":"12480","scopus_import":"1","title":"Approximate message passing with spectral initialization for generalized linear models","intvolume":"      2022","publication_status":"published","article_processing_charge":"Yes (via OA deal)","department":[{"_id":"MaMo"}],"date_created":"2023-02-02T08:31:57Z","file_date_updated":"2023-02-02T08:35:52Z","quality_controlled":"1","article_type":"original","publisher":"IOP Publishing"},{"department":[{"_id":"MaSe"}],"article_processing_charge":"No","date_created":"2021-02-17T17:48:46Z","publication_status":"published","intvolume":"      2021","title":"Disentanglement approach to quantum spin ground states: Field theory and stochastic simulation","_id":"9158","issue":"1","author":[{"id":"42832B76-F248-11E8-B48F-1D18A9856A87","last_name":"De Nicola","first_name":"Stefano","full_name":"De Nicola, Stefano","orcid":"0000-0002-4842-6671"}],"publisher":"IOP Publishing","article_type":"original","ec_funded":1,"quality_controlled":"1","file_date_updated":"2021-02-19T14:04:40Z","day":"05","doi":"10.1088/1742-5468/abc7c7","abstract":[{"lang":"eng","text":"While several tools have been developed to study the ground state of many-body quantum spin systems, the limitations of existing techniques call for the exploration of new approaches. In this manuscript we develop an alternative analytical and numerical framework for many-body quantum spin ground states, based on the disentanglement formalism. In this approach, observables are exactly expressed as Gaussian-weighted functional integrals over scalar fields. We identify the leading contribution to these integrals, given by the saddle point of a suitable effective action. Analytically, we develop a field-theoretical expansion of the functional integrals, performed by means of appropriate Feynman rules. The expansion can be truncated to a desired order to obtain analytical approximations to observables. Numerically, we show that the disentanglement approach can be used to compute ground state expectation values from classical stochastic processes. While the associated fluctuations grow exponentially with imaginary time and the system size, this growth can be mitigated by means of an importance sampling scheme based on knowledge of the saddle point configuration. We illustrate the advantages and limitations of our methods by considering the quantum Ising model in 1, 2 and 3 spatial dimensions. Our analytical and numerical approaches are applicable to a broad class of systems, bridging concepts from quantum lattice models, continuum field theory, and classical stochastic processes."}],"citation":{"chicago":"De Nicola, Stefano. “Disentanglement Approach to Quantum Spin Ground States: Field Theory and Stochastic Simulation.” <i>Journal of Statistical Mechanics: Theory and Experiment</i>. IOP Publishing, 2021. <a href=\"https://doi.org/10.1088/1742-5468/abc7c7\">https://doi.org/10.1088/1742-5468/abc7c7</a>.","ieee":"S. De Nicola, “Disentanglement approach to quantum spin ground states: Field theory and stochastic simulation,” <i>Journal of Statistical Mechanics: Theory and Experiment</i>, vol. 2021, no. 1. IOP Publishing, 2021.","ama":"De Nicola S. Disentanglement approach to quantum spin ground states: Field theory and stochastic simulation. <i>Journal of Statistical Mechanics: Theory and Experiment</i>. 2021;2021(1). doi:<a href=\"https://doi.org/10.1088/1742-5468/abc7c7\">10.1088/1742-5468/abc7c7</a>","apa":"De Nicola, S. (2021). Disentanglement approach to quantum spin ground states: Field theory and stochastic simulation. <i>Journal of Statistical Mechanics: Theory and Experiment</i>. IOP Publishing. <a href=\"https://doi.org/10.1088/1742-5468/abc7c7\">https://doi.org/10.1088/1742-5468/abc7c7</a>","ista":"De Nicola S. 2021. Disentanglement approach to quantum spin ground states: Field theory and stochastic simulation. Journal of Statistical Mechanics: Theory and Experiment. 2021(1), 013101.","mla":"De Nicola, Stefano. “Disentanglement Approach to Quantum Spin Ground States: Field Theory and Stochastic Simulation.” <i>Journal of Statistical Mechanics: Theory and Experiment</i>, vol. 2021, no. 1, 013101, IOP Publishing, 2021, doi:<a href=\"https://doi.org/10.1088/1742-5468/abc7c7\">10.1088/1742-5468/abc7c7</a>.","short":"S. De Nicola, Journal of Statistical Mechanics: Theory and Experiment 2021 (2021)."},"year":"2021","date_updated":"2023-08-07T13:46:28Z","external_id":{"isi":["000605080300001"]},"isi":1,"volume":2021,"acknowledgement":"S D N would like to thank M J Bhaseen, J Chalker, B Doyon, V Gritsev, A Lamacraft,\r\nA Michailidis and M Serbyn for helpful feedback and stimulating conversations. S D N\r\nacknowledges funding from the Institute of Science and Technology (IST) Austria, and\r\nfrom the European Union’s Horizon 2020 research and innovation program under the\r\nMarie Sk\blodowska-Curie Grant Agreement No. 754411. S D N also acknowledges funding\r\nfrom the EPSRC Center for Doctoral Training in Cross-Disciplinary Approaches to Non-\r\nEquilibrium Systems (CANES) under Grant EP/L015854/1. S D N is grateful to IST\r\nAustria for providing open access funding.","ddc":["530"],"project":[{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"oa_version":"Published Version","article_number":"013101","month":"01","has_accepted_license":"1","publication":"Journal of Statistical Mechanics: Theory and Experiment","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability","Statistical and Nonlinear Physics"],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["1742-5468"]},"oa":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"type":"journal_article","date_published":"2021-01-05T00:00:00Z","file":[{"creator":"dernst","file_id":"9172","relation":"main_file","access_level":"open_access","success":1,"file_name":"2021_JourStatMech_deNicola.pdf","content_type":"application/pdf","date_updated":"2021-02-19T14:04:40Z","checksum":"64e2aae4837790db26e1dd1986c69c07","file_size":1693609,"date_created":"2021-02-19T14:04:40Z"}],"status":"public","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8"},{"article_processing_charge":"No","date_created":"2020-09-17T10:42:22Z","publication_status":"published","intvolume":"       188","title":"On the local Birkhoff conjecture for convex billiards","_id":"8421","issue":"1","author":[{"first_name":"Vadim","last_name":"Kaloshin","orcid":"0000-0002-6051-2628","full_name":"Kaloshin, Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425"},{"first_name":"Alfonso","last_name":"Sorrentino","full_name":"Sorrentino, Alfonso"}],"publisher":"Annals of Mathematics, Princeton U","article_type":"original","quality_controlled":"1","page":"315-380","day":"01","doi":"10.4007/annals.2018.188.1.6","arxiv":1,"abstract":[{"text":"The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.","lang":"eng"}],"citation":{"ieee":"V. Kaloshin and A. Sorrentino, “On the local Birkhoff conjecture for convex billiards,” <i>Annals of Mathematics</i>, vol. 188, no. 1. Annals of Mathematics, Princeton U, pp. 315–380, 2018.","chicago":"Kaloshin, Vadim, and Alfonso Sorrentino. “On the Local Birkhoff Conjecture for Convex Billiards.” <i>Annals of Mathematics</i>. Annals of Mathematics, Princeton U, 2018. <a href=\"https://doi.org/10.4007/annals.2018.188.1.6\">https://doi.org/10.4007/annals.2018.188.1.6</a>.","ama":"Kaloshin V, Sorrentino A. On the local Birkhoff conjecture for convex billiards. <i>Annals of Mathematics</i>. 2018;188(1):315-380. doi:<a href=\"https://doi.org/10.4007/annals.2018.188.1.6\">10.4007/annals.2018.188.1.6</a>","apa":"Kaloshin, V., &#38; Sorrentino, A. (2018). On the local Birkhoff conjecture for convex billiards. <i>Annals of Mathematics</i>. Annals of Mathematics, Princeton U. <a href=\"https://doi.org/10.4007/annals.2018.188.1.6\">https://doi.org/10.4007/annals.2018.188.1.6</a>","ista":"Kaloshin V, Sorrentino A. 2018. On the local Birkhoff conjecture for convex billiards. Annals of Mathematics. 188(1), 315–380.","short":"V. Kaloshin, A. Sorrentino, Annals of Mathematics 188 (2018) 315–380.","mla":"Kaloshin, Vadim, and Alfonso Sorrentino. “On the Local Birkhoff Conjecture for Convex Billiards.” <i>Annals of Mathematics</i>, vol. 188, no. 1, Annals of Mathematics, Princeton U, 2018, pp. 315–80, doi:<a href=\"https://doi.org/10.4007/annals.2018.188.1.6\">10.4007/annals.2018.188.1.6</a>."},"year":"2018","date_updated":"2021-01-12T08:19:10Z","external_id":{"arxiv":["1612.09194"]},"volume":188,"extern":"1","oa_version":"Preprint","month":"07","publication":"Annals of Mathematics","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["0003-486X"]},"oa":1,"type":"journal_article","date_published":"2018-07-01T00:00:00Z","main_file_link":[{"url":"https://arxiv.org/abs/1612.09194","open_access":"1"}],"status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"article_type":"original","publisher":"JSTOR","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"language":[{"iso":"eng"}],"quality_controlled":"1","page":"729-741","intvolume":"       150","title":"An extension of the Artin-Mazur theorem","month":"09","article_processing_charge":"No","date_created":"2020-09-18T10:50:28Z","oa_version":"None","publication_status":"published","issue":"2","author":[{"last_name":"Kaloshin","first_name":"Vadim","full_name":"Kaloshin, Vadim","orcid":"0000-0002-6051-2628","id":"FE553552-CDE8-11E9-B324-C0EBE5697425"}],"_id":"8526","publication":"The Annals of Mathematics","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","extern":"1","volume":150,"publication_identifier":{"issn":["0003-486X"]},"day":"01","doi":"10.2307/121093","type":"journal_article","date_published":"1999-09-01T00:00:00Z","citation":{"ista":"Kaloshin V. 1999. An extension of the Artin-Mazur theorem. The Annals of Mathematics. 150(2), 729–741.","short":"V. Kaloshin, The Annals of Mathematics 150 (1999) 729–741.","mla":"Kaloshin, Vadim. “An Extension of the Artin-Mazur Theorem.” <i>The Annals of Mathematics</i>, vol. 150, no. 2, JSTOR, 1999, pp. 729–41, doi:<a href=\"https://doi.org/10.2307/121093\">10.2307/121093</a>.","chicago":"Kaloshin, Vadim. “An Extension of the Artin-Mazur Theorem.” <i>The Annals of Mathematics</i>. JSTOR, 1999. <a href=\"https://doi.org/10.2307/121093\">https://doi.org/10.2307/121093</a>.","ieee":"V. Kaloshin, “An extension of the Artin-Mazur theorem,” <i>The Annals of Mathematics</i>, vol. 150, no. 2. JSTOR, pp. 729–741, 1999.","apa":"Kaloshin, V. (1999). An extension of the Artin-Mazur theorem. <i>The Annals of Mathematics</i>. JSTOR. <a href=\"https://doi.org/10.2307/121093\">https://doi.org/10.2307/121093</a>","ama":"Kaloshin V. An extension of the Artin-Mazur theorem. <i>The Annals of Mathematics</i>. 1999;150(2):729-741. doi:<a href=\"https://doi.org/10.2307/121093\">10.2307/121093</a>"},"year":"1999","date_updated":"2021-01-12T08:19:53Z"}]