{"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/backpropagation-through-back-substitution","title":"Backpropagation through Back Substitution with a Backslash","arxiv_id":"2303.15449","date":"2023-02-23","proceeding":null,"authors":["Alan Edelman","Ekin Akyurek","Yuyang Wang"],"abstract":"We present a linear algebra formulation of backpropagation which allows the calculation of gradients by using a generically written ``backslash'' or Gaussian elimination on triangular systems of equations. Generally, the matrix elements are operators. This paper has three contributions: (i) it is of intellectual value to replace traditional treatments of automatic differentiation with a (left acting) operator theoretic, graph-based approach; (ii) operators can be readily placed in matrices in software in programming languages such as Julia as an implementation option; (iii) we introduce a novel notation, ``transpose dot'' operator ``$\\{\\}^{T_\\bullet}$'' that allows for the reversal of operators. We further demonstrate the elegance of the operators approach in a suitable programming language consisting of generic linear algebra operators such as Julia \\cite{bezanson2017julia}, and that it is possible to realize this abstraction in code. Our implementation shows how generic linear algebra can allow operators as elements of matrices. In contrast to ``operator overloading,'' where backslash would normally have to be rewritten to take advantage of operators, with ``generic programming'' there is no such need.","url_abs":"https://arxiv.org/abs/2303.15449v2","url_pdf":"https://arxiv.org/pdf/2303.15449v2.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":"backpropagation-through-back-substitution","repo_url":"https://github.com/alanedelman/backbackback","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}