{"paper":{"title":"Error Bound of Empirical $\\ell_2$ Risk Minimization for Noisy Standard and Generalized Phase Retrieval Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Junren Chen, Michael K. Ng","submitted_at":"2022-05-27T08:38:33Z","abstract_excerpt":"In this paper, we study the estimation performance of empirical $\\ell_2$ risk minimization (ERM) in noisy (standard) phase retrieval (NPR) given by $y_k = |\\alpha_k^*x_0|^2+\\eta_k$, or noisy generalized phase retrieval (NGPR) formulated as $y_k = x_0^*A_kx_0 + \\eta_k$, where $x_0\\in\\mathbb{K}^d$ is the desired signal, $n$ is the sample size, $\\eta= (\\eta_1,...,\\eta_n)^\\top$ is the noise vector. We establish new error bounds under different noise patterns, and our proofs are valid for both $\\mathbb{K}=\\mathbb{R}$ and $\\mathbb{K}=\\mathbb{C}$. In NPR under arbitrary noise vector $\\eta$, we derive"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.13827","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2205.13827/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}