{"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/acdc-a-structured-efficient-linear-layer","title":"ACDC: A Structured Efficient Linear Layer","arxiv_id":"1511.05946","date":"2015-11-18","proceeding":null,"authors":["Marcin Moczulski","Misha Denil","Jeremy Appleyard","Nando de Freitas"],"abstract":"The linear layer is one of the most pervasive modules in deep learning\nrepresentations. However, it requires $O(N^2)$ parameters and $O(N^2)$\noperations. These costs can be prohibitive in mobile applications or prevent\nscaling in many domains. Here, we introduce a deep, differentiable,\nfully-connected neural network module composed of diagonal matrices of\nparameters, $\\mathbf{A}$ and $\\mathbf{D}$, and the discrete cosine transform\n$\\mathbf{C}$. The core module, structured as $\\mathbf{ACDC^{-1}}$, has $O(N)$\nparameters and incurs $O(N log N )$ operations. We present theoretical results\nshowing how deep cascades of ACDC layers approximate linear layers. ACDC is,\nhowever, a stand-alone module and can be used in combination with any other\ntypes of module. In our experiments, we show that it can indeed be successfully\ninterleaved with ReLU modules in convolutional neural networks for image\nrecognition. Our experiments also study critical factors in the training of\nthese structured modules, including initialization and depth. Finally, this\npaper also provides a connection between structured linear transforms used in\ndeep learning and the field of Fourier optics, illustrating how ACDC could in\nprinciple be implemented with lenses and diffractive elements.","url_abs":"http://arxiv.org/abs/1511.05946v5","url_pdf":"http://arxiv.org/pdf/1511.05946v5.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":"acdc-a-structured-efficient-linear-layer","repo_url":"https://github.com/mdenil/acdc-torch","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"torch","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"acdc-a-structured-efficient-linear-layer","repo_url":"https://github.com/gngdb/pytorch-acdc","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[{"method_slug":"linear-layer","method_name":"Linear Layer"},{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1511.05946","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1511.05946"}},"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/mdenil/acdc-torch","reach":{"status":"ok","spdx":"MIT"}},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/gngdb/pytorch-acdc","reach":null}],"summary":{"ran_honours":2},"by_repo_kind":{"listed":{"samples":2,"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":"4ed80429e85bef65","entry":"count_params","repo":"gngdb/pytorch-acdc","repo_kind":"listed","path":"linear_layer_approx.py","file_url":"https://github.com/gngdb/pytorch-acdc/blob/HEAD/linear_layer_approx.py","link_basis":"first_harvest_node","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":"4ed80429e85bef65"}},{"code_sha256_prefix":"0ec5de7410b5a599","entry":"sample_experiment","repo":"gngdb/pytorch-acdc","repo_kind":"listed","path":"linear_layer_approx.py","file_url":"https://github.com/gngdb/pytorch-acdc/blob/HEAD/linear_layer_approx.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"well_formed","behaviour_fingerprint":true,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"0ec5de7410b5a599"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}