{"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/dense-and-sparse-coding-theory-and","title":"Towards improving discriminative reconstruction via simultaneous dense and sparse coding","arxiv_id":"2006.09534","date":"2020-06-16","proceeding":null,"authors":["Abiy Tasissa","Emmanouil Theodosis","Bahareh Tolooshams","Demba Ba"],"abstract":"Discriminative features extracted from the sparse coding model have been shown to perform well for classification. Recent deep learning architectures have further improved reconstruction in inverse problems by considering new dense priors learned from data. We propose a novel dense and sparse coding model that integrates both representation capability and discriminative features. The model studies the problem of recovering a dense vector $\\mathbf{x}$ and a sparse vector $\\mathbf{u}$ given measurements of the form $\\mathbf{y} = \\mathbf{A}\\mathbf{x}+\\mathbf{B}\\mathbf{u}$. Our first analysis proposes a geometric condition based on the minimal angle between spanning subspaces corresponding to the matrices $\\mathbf{A}$ and $\\mathbf{B}$ that guarantees unique solution to the model. The second analysis shows that, under mild assumptions, a convex program recovers the dense and sparse components. We validate the effectiveness of the model on simulated data and propose a dense and sparse autoencoder (DenSaE) tailored to learning the dictionaries from the dense and sparse model. We demonstrate that (i) DenSaE denoises natural images better than architectures derived from the sparse coding model ($\\mathbf{B}\\mathbf{u}$), (ii) in the presence of noise, training the biases in the latter amounts to implicitly learning the $\\mathbf{A}\\mathbf{x} + \\mathbf{B}\\mathbf{u}$ model, (iii) $\\mathbf{A}$ and $\\mathbf{B}$ capture low- and high-frequency contents, respectively, and (iv) compared to the sparse coding model, DenSaE offers a balance between discriminative power and representation.","url_abs":"https://arxiv.org/abs/2006.09534v3","url_pdf":"https://arxiv.org/pdf/2006.09534v3.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":"dense-and-sparse-coding-theory-and","repo_url":"https://github.com/btolooshams/densae","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"compressive-sensing","task_name":"Compressive Sensing"},{"task_slug":"dictionary-learning","task_name":"Dictionary Learning"}],"methods":[{"method_slug":"sparse-autoencoder","method_name":"Sparse Autoencoder"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2006.09534","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2006.09534"}},"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. 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