{"paper":{"title":"Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Anherutowa Calvo","submitted_at":"2026-05-22T23:31:38Z","abstract_excerpt":"The curvature exponent $\\alpha$ in $h_k \\propto \\sigma_k^\\alpha$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($\\alpha \\approx 2$ for convolutions, $\\approx 1$ for transformer attention, $< 1$ for MLP up-projections). Why? We prove the Spectral Alignment Decomposition: $\\alpha = 2 + d\\log\\Phi_k / d\\log\\sigma_k$, where $\\Phi_k$ measures alignment between Kronecker factor eigenbases and gradient singular directions. This reduces \"why does $\\alpha$ vary?\" to a geometric question we answer for LayerNorm, residual connections, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02596","kind":"arxiv","version":1},"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/2606.02596/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"}