{"paper":{"title":"On approximating $\\nabla f$ with neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Saeed Saremi","submitted_at":"2019-10-28T15:12:42Z","abstract_excerpt":"Consider a feedforward neural network $\\psi: \\mathbb{R}^d\\rightarrow \\mathbb{R}^d$ such that $\\psi\\approx \\nabla f$, where $f:\\mathbb{R}^d \\rightarrow \\mathbb{R}$ is a smooth function, therefore $\\psi$ must satisfy $\\partial_j \\psi_i = \\partial_i \\psi_j$ pointwise. We prove a theorem that a $\\psi$ network with more than one hidden layer can only represent one feature in its first hidden layer; this is a dramatic departure from the well-known results for one hidden layer. The proof of the theorem is straightforward, where two backward paths and a weight-tying matrix play the key roles. We then "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.12744","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/1910.12744/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"}