{"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/o-k-equivariant-dimensionality-reduction-on","title":"$O(k)$-Equivariant Dimensionality Reduction on Stiefel Manifolds","arxiv_id":"2309.10775","date":"2023-09-19","proceeding":null,"authors":["Andrew Lee","Harlin Lee","Jose A. Perea","Nikolas Schonsheck","Madeleine Weinstein"],"abstract":"Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifolds, $V_k(\\mathbb{R}^N)$ and $Gr(k, \\mathbb{R}^N)$ respectively, and benefit from projection onto lower-dimensional Stiefel and Grassmannian manifolds. In this work, we propose an algorithm called \\textit{Principal Stiefel Coordinates (PSC)} to reduce data dimensionality from $ V_k(\\mathbb{R}^N)$ to $V_k(\\mathbb{R}^n)$ in an \\textit{$O(k)$-equivariant} manner ($k \\leq n \\ll N$). We begin by observing that each element $\\alpha \\in V_n(\\mathbb{R}^N)$ defines an isometric embedding of $V_k(\\mathbb{R}^n)$ into $V_k(\\mathbb{R}^N)$. Next, we describe two ways of finding a suitable embedding map $\\alpha$: one via an extension of principal component analysis ($\\alpha_{PCA}$), and one that further minimizes data fit error using gradient descent ($\\alpha_{GD}$). Then, we define a continuous and $O(k)$-equivariant map $\\pi_\\alpha$ that acts as a \"closest point operator\" to project the data onto the image of $V_k(\\mathbb{R}^n)$ in $V_k(\\mathbb{R}^N)$ under the embedding determined by $\\alpha$, while minimizing distortion. Because this dimensionality reduction is $O(k)$-equivariant, these results extend to Grassmannian manifolds as well. Lastly, we show that $\\pi_{\\alpha_{PCA}}$ globally minimizes projection error in a noiseless setting, while $\\pi_{\\alpha_{GD}}$ achieves a meaningfully different and improved outcome when the data does not lie exactly on the image of a linearly embedded lower-dimensional Stiefel manifold as above. Multiple numerical experiments using synthetic and real-world data are performed.","url_abs":"https://arxiv.org/abs/2309.10775v3","url_pdf":"https://arxiv.org/pdf/2309.10775v3.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":"o-k-equivariant-dimensionality-reduction-on","repo_url":"https://github.com/crispfish/psc","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"o-k-equivariant-dimensionality-reduction-on","repo_url":"https://github.com/harlinlee/psc","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"dimensionality-reduction","task_name":"Dimensionality Reduction"}],"methods":[{"method_slug":"pca","method_name":"PCA"}],"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}