{"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/on-the-approximation-capabilities-of-relu","title":"On the Approximation Properties of Random ReLU Features","arxiv_id":"1810.04374","date":"2018-10-10","proceeding":null,"authors":["Yitong Sun","Anna Gilbert","Ambuj Tewari"],"abstract":"We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features method is a universally consistent learning algorithm. We prove that despite the universality of the RKHS induced by the random ReLU features, composition of functions in it generates substantially more complicated functions that are harder to approximate than those functions simply in the RKHS. We also prove that such composite functions can be efficiently approximated by multi-layer ReLU networks with bounded weights. This depth separation result shows that the random ReLU features models suffer from the same weakness as that of shallow models. We show in experiments that the performance of random ReLU features is comparable to that of random Fourier features and, in general, has a lower computational cost. We also demonstrate that when the target function is the composite function as described in the depth separation theorem, 3-layer neural networks indeed outperform both random ReLU features and 2-layer neural networks.","url_abs":"https://arxiv.org/abs/1810.04374v3","url_pdf":"https://arxiv.org/pdf/1810.04374v3.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":"on-the-approximation-capabilities-of-relu","repo_url":"https://github.com/syitong/randrelu","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"tf","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1810.04374","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1810.04374"}},"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/syitong/randrelu","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"ran_draft_wrong":1,"ran_honours":1,"unverified":9},"by_repo_kind":{"official":{"samples":11,"ran":2,"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":"4d56f8ece9741bf8","entry":"get_train_test_data","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/libmnist.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/libmnist.py","link_basis":"plan_row","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":"deterministic","behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"4d56f8ece9741bf8"}},{"code_sha256_prefix":"39db295096d110b1","entry":"read_MNIST_data","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/libmnist.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/libmnist.py","link_basis":"plan_row","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"well_formed","behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"39db295096d110b1"}},{"code_sha256_prefix":"289b03842b400880","entry":"clip_by_maxnorm","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/librf.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/librf.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":"289b03842b400880"}},{"code_sha256_prefix":"3f03fa3888a0173c","entry":"gamma_est","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/librf.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/librf.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":"3f03fa3888a0173c"}},{"code_sha256_prefix":"d6b9558bc0e2c5a9","entry":"mkdir","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/lib/utils.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/lib/utils.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":"d6b9558bc0e2c5a9"}},{"code_sha256_prefix":"f46af98a4d946ad1","entry":"mollify","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/generate_data.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/generate_data.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":"f46af98a4d946ad1"}},{"code_sha256_prefix":"cf5a764013dc7b9e","entry":"print_params","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/result_show.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/result_show.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":"cf5a764013dc7b9e"}},{"code_sha256_prefix":"1484361003ebf8c8","entry":"print_test_results","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/result_show.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/result_show.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":"1484361003ebf8c8"}},{"code_sha256_prefix":"fcfe2c47d08ce41c","entry":"read_params","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/experiments.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/experiments.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":"fcfe2c47d08ce41c"}},{"code_sha256_prefix":"880bb8bd1dc9f8c3","entry":"validate","repo":"syitong/randrelu","repo_kind":"official","path":"depth_sep/experiments.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/depth_sep/experiments.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":"880bb8bd1dc9f8c3"}},{"code_sha256_prefix":"c0b8a1ed11eed7c1","entry":"validate","repo":"syitong/randrelu","repo_kind":"official","path":"pycode/experiments.py","file_url":"https://github.com/syitong/randrelu/blob/HEAD/pycode/experiments.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":"c0b8a1ed11eed7c1"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}