{"paper":{"title":"Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"In high-dimensional ERM with non-Gaussian data, the estimator's projection on a test point follows the convolution of a generally non-Gaussian distribution with an independent Gaussian whose variance is set by the trace of the estimator's 2","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Chiheb Yaakoubi, Cosme Louart, Malik Tiomoko, Zhenyu Liao","submitted_at":"2026-04-03T16:07:02Z","abstract_excerpt":"We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min-Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min-max characterization of key statistics, enabling approximation of the mean $\\mu_{\\hat{\\theta}}$ and covariance $C_{\\hat{\\theta}}$ of the ERM estimator $\\hat{\\theta}$.\n  Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training dat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate x independent of the training data, the projection θ̂⊤x approximately follows the convolution of the (generally non-Gaussian) distribution of μ_θ̂⊤x with an independent centered Gaussian variable of variance Tr(C_θ̂ E[xx⊤])","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"the heuristic extension of the Convex Gaussian Min-Max Theorem to non-Gaussian settings under a concentration assumption on the data matrix","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In high-dimensional convex ERM with non-Gaussian data, the projection of the estimator onto a test covariate asymptotically follows the convolution of a generally non-Gaussian term with an independent centered Gaussian whose variance is the trace of the estimator covariance times the data second-mom","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"In high-dimensional ERM with non-Gaussian data, the estimator's projection on a test point follows the convolution of a generally non-Gaussian distribution with an independent Gaussian whose variance is set by the trace of the estimator's 2","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"45da8ea5e864f82801169551c28b2f9e4a77c0347aea89b7190e96f591fd0d21"},"source":{"id":"2604.03146","kind":"arxiv","version":2},"verdict":{"id":"38f6bd0f-fefd-42f4-928b-7b41d3204007","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T18:25:32.317025Z","strongest_claim":"under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate x independent of the training data, the projection θ̂⊤x approximately follows the convolution of the (generally non-Gaussian) distribution of μ_θ̂⊤x with an independent centered Gaussian variable of variance Tr(C_θ̂ E[xx⊤])","one_line_summary":"In high-dimensional convex ERM with non-Gaussian data, the projection of the estimator onto a test covariate asymptotically follows the convolution of a generally non-Gaussian term with an independent centered Gaussian whose variance is the trace of the estimator covariance times the data second-mom","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"the heuristic extension of the Convex Gaussian Min-Max Theorem to non-Gaussian settings under a concentration assumption on the data matrix","pith_extraction_headline":"In high-dimensional ERM with non-Gaussian data, the estimator's projection on a test point follows the convolution of a generally non-Gaussian distribution with an independent Gaussian whose variance is set by the trace of the estimator's 2"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.03146/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":2,"snapshot_sha256":"b42a284ed68de677ce3bee9f3dbbb493bf94deeed93729e420a29fc084c61745"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}