{"ID":2860588,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.03809","arxiv_id":"2510.03809","title":"Spectral Thresholds for Identifiability and Stability:Finite-Sample Phase Transitions in High-Dimensional Learning","abstract":"In high-dimensional learning, models remain stable until they collapse abruptly once the sample size falls below a critical level. This instability is not algorithm-specific but a geometric mechanism: when the weakest Fisher eigendirection falls beneath sample-level fluctuations, identifiability fails. Our Fisher Threshold Theorem formalizes this by proving that stability requires the minimal Fisher eigenvalue to exceed an explicit $O(\\sqrt{d/n})$ bound. Unlike prior asymptotic or model-specific criteria, this threshold is finite-sample and necessary, marking a sharp phase transition between reliable concentration and inevitable failure. To make the principle constructive, we introduce the Fisher floor, a verifiable spectral regularization robust to smoothing and preconditioning. Synthetic experiments on Gaussian mixtures and logistic models confirm the predicted transition, consistent with $d/n$ scaling. Statistically, the threshold sharpens classical eigenvalue conditions into a non-asymptotic law; learning-theoretically, it defines a spectral sample-complexity frontier, bridging theory with diagnostics for robust high-dimensional inference.","short_abstract":"In high-dimensional learning, models remain stable until they collapse abruptly once the sample size falls below a critical level. This instability is not algorithm-specific but a geometric mechanism: when the weakest Fisher eigendirection falls beneath sample-level fluctuations, identifiability fails. Our Fisher Thres...","url_abs":"https://arxiv.org/abs/2510.03809","url_pdf":"https://arxiv.org/pdf/2510.03809v2","authors":"[\"William Hao-Cheng Huang\"]","published":"2025-10-04T13:33:48Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}