{"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/cross-validation-in-high-dimensional-spaces-a","title":"Cross-validation in high-dimensional spaces: a lifeline for least-squares models and multi-class LDA","arxiv_id":"1803.10016","date":"2018-03-27","proceeding":null,"authors":["Matthias S. Treder"],"abstract":"Least-squares models such as linear regression and Linear Discriminant\nAnalysis (LDA) are amongst the most popular statistical learning techniques.\nHowever, since their computation time increases cubically with the number of\nfeatures, they are inefficient in high-dimensional neuroimaging datasets.\nFortunately, for k-fold cross-validation, an analytical approach has been\ndeveloped that yields the exact cross-validated predictions in least-squares\nmodels without explicitly training the model. Its computation time grows with\nthe number of test samples. Here, this approach is systematically investigated\nin the context of cross-validation and permutation testing. LDA is used\nexemplarily but results hold for all other least-squares methods. Furthermore,\na non-trivial extension to multi-class LDA is formally derived. The analytical\napproach is evaluated using complexity calculations, simulations, and\npermutation testing of an EEG/MEG dataset. Depending on the ratio between\nfeatures and samples, the analytical approach is up to 10,000x faster than the\nstandard approach (retraining the model on each training set). This allows for\na fast cross-validation of least-squares models and multi-class LDA in\nhigh-dimensional data, with obvious applications in multi-dimensional datasets,\nRepresentational Similarity Analysis, and permutation testing.","url_abs":"http://arxiv.org/abs/1803.10016v1","url_pdf":"http://arxiv.org/pdf/1803.10016v1.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":"cross-validation-in-high-dimensional-spaces-a","repo_url":"https://github.com/treder/Fast-Least-Squares","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"eeg-1","task_name":"EEG"},{"task_slug":"eeg","task_name":"Electroencephalogram (EEG)"}],"methods":[{"method_slug":"lda","method_name":"LDA"},{"method_slug":"linear-regression","method_name":"Linear Regression"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}