{"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/training-deep-networks-with-structured-layers","title":"Training Deep Networks with Structured Layers by Matrix Backpropagation","arxiv_id":"1509.07838","date":"2015-09-25","proceeding":null,"authors":["Catalin Ionescu","Orestis Vantzos","Cristian Sminchisescu"],"abstract":"Deep neural network architectures have recently produced excellent results in\na variety of areas in artificial intelligence and visual recognition, well\nsurpassing traditional shallow architectures trained using hand-designed\nfeatures. The power of deep networks stems both from their ability to perform\nlocal computations followed by pointwise non-linearities over increasingly\nlarger receptive fields, and from the simplicity and scalability of the\ngradient-descent training procedure based on backpropagation. An open problem\nis the inclusion of layers that perform global, structured matrix computations\nlike segmentation (e.g. normalized cuts) or higher-order pooling (e.g.\nlog-tangent space metrics defined over the manifold of symmetric positive\ndefinite matrices) while preserving the validity and efficiency of an\nend-to-end deep training framework. In this paper we propose a sound\nmathematical apparatus to formally integrate global structured computation into\ndeep computation architectures. At the heart of our methodology is the\ndevelopment of the theory and practice of backpropagation that generalizes to\nthe calculus of adjoint matrix variations. The proposed matrix backpropagation\nmethodology applies broadly to a variety of problems in machine learning or\ncomputational perception. Here we illustrate it by performing visual\nsegmentation experiments using the BSDS and MSCOCO benchmarks, where we show\nthat deep networks relying on second-order pooling and normalized cuts layers,\ntrained end-to-end using matrix backpropagation, outperform counterparts that\ndo not take advantage of such global layers.","url_abs":"http://arxiv.org/abs/1509.07838v4","url_pdf":"http://arxiv.org/pdf/1509.07838v4.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":"training-deep-networks-with-structured-layers","repo_url":"https://github.com/Abdelpakey/SVD-solution-for-the-ill-posed","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"torch","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1509.07838","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}