{"paper":{"title":"Landscape analysis for shallow neural networks: Complete classification of critical points for cubic activation and affine target functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Diyora Salimova, Ilkhom Mukhammadiev, Shokhrukh Ibragimov","submitted_at":"2026-07-16T16:22:16Z","abstract_excerpt":"In this paper, we study the optimization landscape induced by the true loss for shallow polynomial neural networks (PNNs) with $\\mathfrak{h} \\in \\mathbb{N}$ neurons on the hidden layer, one-dimensional input and output layers, and a monomial activation of degree $d \\in \\mathbb{N}$, trained against a non-constant affine linear target function. Our first main result provides for arbitrary activation degree $d$ a sharp existence/non-existence criterion for \\emph{global minimizers} with necessary structural conditions. We show that the infimum of the loss is always zero and achievable with at leas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15173","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/2607.15173/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"}