{"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/identifiability-in-exact-multilayer-sparse","title":"Efficient Identification of Butterfly Sparse Matrix Factorizations","arxiv_id":"2110.01230","date":"2021-10-04","proceeding":null,"authors":["Léon Zheng","Elisa Riccietti","Rémi Gribonval"],"abstract":"Fast transforms correspond to factorizations of the form $\\mathbf{Z} = \\mathbf{X}^{(1)} \\ldots \\mathbf{X}^{(J)}$, where each factor $ \\mathbf{X}^{(\\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any $N \\times N$ matrix having the so-called butterfly structure admits an essentially unique factorization into $J$ butterfly factors (where $N = 2^{J}$), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size $N=2^J$. Computing such factorizations costs $\\mathcal{O}(N^{2})$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\\mathcal{O}(N \\log N)$ matrix-vector multiplications and have the potential to be applied to compress deep neural networks.","url_abs":"https://arxiv.org/abs/2110.01230v9","url_pdf":"https://arxiv.org/pdf/2110.01230v9.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":"identifiability-in-exact-multilayer-sparse","repo_url":"https://github.com/leonzheng2/efficient-butterfly","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"discrete-cosine-transform","method_name":"Discrete Cosine Transform"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}