{"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/on-the-reconstruction-risk-of-convolutional","title":"On the Reconstruction Risk of Convolutional Sparse Dictionary Learning","arxiv_id":"1708.08587","date":"2017-08-29","proceeding":null,"authors":["Shashank Singh","Barnabás Póczos","Jian Ma"],"abstract":"Sparse dictionary learning (SDL) has become a popular method for adaptively\nidentifying parsimonious representations of a dataset, a fundamental problem in\nmachine learning and signal processing. While most work on SDL assumes a\ntraining dataset of independent and identically distributed samples, a variant\nknown as convolutional sparse dictionary learning (CSDL) relaxes this\nassumption, allowing more general sequential data sources, such as time series\nor other dependent data. Although recent work has explored the statistical\nproperties of classical SDL, the statistical properties of CSDL remain\nunstudied. This paper begins to study this by identifying the minimax\nconvergence rate of CSDL in terms of reconstruction risk, by both upper\nbounding the risk of an established CSDL estimator and proving a matching\ninformation-theoretic lower bound. Our results indicate that consistency in\nreconstruction risk is possible precisely in the `ultra-sparse' setting, in\nwhich the sparsity (i.e., the number of feature occurrences) is in $o(N)$ in\nterms of the length N of the training sequence. Notably, our results make very\nweak assumptions, allowing arbitrary dictionaries and dependent measurement\nnoise. Finally, we verify our theoretical results with numerical experiments on\nsynthetic data.","url_abs":"http://arxiv.org/abs/1708.08587v2","url_pdf":"http://arxiv.org/pdf/1708.08587v2.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":"on-the-reconstruction-risk-of-convolutional","repo_url":"https://github.com/sss1/convolutional-dictionary","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"dictionary-learning","task_name":"Dictionary Learning"},{"task_slug":"time-series-1","task_name":"Time Series"},{"task_slug":"time-series","task_name":"Time Series Analysis"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}