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In this work we combine the unique geometric structure of the sparse principal component analysis problem with recent advances in convex optimization to develop novel gradient-based sparse principal component analysis algorithms. These algorithms enjoy the same global convergence guarantee as the original alternating direction method of multipliers, and can be more efficiently implemented with the rich toolbox developed for gradient methods from the deep learning literature. Most notably, these gradient-based algorithms can be combined with stochastic gradient descent methods to produce efficient online sparse principal component analysis algorithms with provable numerical and statistical performance guarantees. The practical performance and usefulness of the new algorithms are demonstrated in various simulation studies. As an application, we show how the scalability and statistical accuracy of our method enable us to find interesting functional gene groups in high-dimensional RNA sequencing data.<\/jats:p>","DOI":"10.1093\/biomet\/asac041","type":"journal-article","created":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T10:06:12Z","timestamp":1657620372000},"page":"339-360","source":"Crossref","is-referenced-by-count":6,"title":["Gradient-based sparse principal component analysis with extensions to online learning"],"prefix":"10.1093","volume":"110","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0109-6692","authenticated-orcid":false,"given":"Yixuan","family":"Qiu","sequence":"first","affiliation":[{"name":"Shanghai University of Finance and Economics School of Statistics and Management, , 777 Guoding Road, Shanghai 200433, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3104-9387","authenticated-orcid":false,"given":"Jing","family":"Lei","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University Department of Statistics and Data Science, , 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, U.S.A"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kathryn","family":"Roeder","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University Department of Statistics and Data Science, , 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, U.S.A"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2022,7,12]]},"reference":[{"key":"2023071200315980500_B1","first-page":"415","article-title":"Optimal strategies and minimax lower bounds for online convex games","author":"Abernethy,","year":"2008"},{"key":"2023071200315980500_B2","first-page":"861","article-title":"Stochastic optimization for PCA and PLS","author":"Arora,","year":"2012"},{"key":"2023071200315980500_B3","doi-asserted-by":"crossref","DOI":"10.1007\/s10107-011-0472-0","article-title":"Incremental proximal methods for large scale convex optimization","volume":"129","author":"Bertsekas,","year":"2011","journal-title":"Math. 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