{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-random-matrix-approach-to-neural-networks","title":"A Random Matrix Approach to Neural Networks","arxiv_id":"1702.05419","date":"2017-02-17","proceeding":null,"authors":["Cosme Louart","Zhenyu Liao","Romain Couillet"],"abstract":"This article studies the Gram random matrix model $G=\\frac1T\\Sigma^{\\rm\nT}\\Sigma$, $\\Sigma=\\sigma(WX)$, classically found in the analysis of random\nfeature maps and random neural networks, where $X=[x_1,\\ldots,x_T]\\in{\\mathbb\nR}^{p\\times T}$ is a (data) matrix of bounded norm, $W\\in{\\mathbb R}^{n\\times\np}$ is a matrix of independent zero-mean unit variance entries, and\n$\\sigma:{\\mathbb R}\\to{\\mathbb R}$ is a Lipschitz continuous (activation)\nfunction --- $\\sigma(WX)$ being understood entry-wise. By means of a key\nconcentration of measure lemma arising from non-asymptotic random matrix\narguments, we prove that, as $n,p,T$ grow large at the same rate, the resolvent\n$Q=(G+\\gamma I_T)^{-1}$, for $\\gamma>0$, has a similar behavior as that met in\nsample covariance matrix models, involving notably the moment\n$\\Phi=\\frac{T}n{\\mathbb E}[G]$, which provides in passing a deterministic\nequivalent for the empirical spectral measure of $G$. Application-wise, this\nresult enables the estimation of the asymptotic performance of single-layer\nrandom neural networks. This in turn provides practical insights into the\nunderlying mechanisms into play in random neural networks, entailing several\nunexpected consequences, as well as a fast practical means to tune the network\nhyperparameters.","url_abs":"http://arxiv.org/abs/1702.05419v2","url_pdf":"http://arxiv.org/pdf/1702.05419v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"a-random-matrix-approach-to-neural-networks","repo_url":"https://github.com/Zhenyu-LIAO/RMT4ELM","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"lemma","task_name":"LEMMA"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1702.05419","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1702.05419"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/Zhenyu-LIAO/RMT4ELM","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"unverified":1},"by_repo_kind":{"official":{"samples":1,"ran":0,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":0,"samples":[{"code_sha256_prefix":"20f3709c60695259","entry":"gen_sig","repo":"Zhenyu-LIAO/RMT4ELM","repo_kind":"official","path":"RMT4ELM.py","file_url":"https://github.com/Zhenyu-LIAO/RMT4ELM/blob/HEAD/RMT4ELM.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"20f3709c60695259"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}