{"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/two-way-kernel-matrix-puncturing-towards","title":"Two-way kernel matrix puncturing: towards resource-efficient PCA and spectral clustering","arxiv_id":"2102.12293","date":"2021-02-24","proceeding":null,"authors":["Romain Couillet","Florent Chatelain","Nicolas Le Bihan"],"abstract":"The article introduces an elementary cost and storage reduction method for spectral clustering and principal component analysis. The method consists in randomly \"puncturing\" both the data matrix $X\\in\\mathbb{C}^{p\\times n}$ (or $\\mathbb{R}^{p\\times n}$) and its corresponding kernel (Gram) matrix $K$ through Bernoulli masks: $S\\in\\{0,1\\}^{p\\times n}$ for $X$ and $B\\in\\{0,1\\}^{n\\times n}$ for $K$. The resulting \"two-way punctured\" kernel is thus given by $K=\\frac{1}{p}[(X \\odot S)^{\\sf H} (X \\odot S)] \\odot B$. We demonstrate that, for $X$ composed of independent columns drawn from a Gaussian mixture model, as $n,p\\to\\infty$ with $p/n\\to c_0\\in(0,\\infty)$, the spectral behavior of $K$ -- its limiting eigenvalue distribution, as well as its isolated eigenvalues and eigenvectors -- is fully tractable and exhibits a series of counter-intuitive phenomena. We notably prove, and empirically confirm on GAN-generated image databases, that it is possible to drastically puncture the data, thereby providing possibly huge computational and storage gains, for a virtually constant (clustering of PCA) performance. This preliminary study opens as such the path towards rethinking, from a large dimensional standpoint, computational and storage costs in elementary machine learning models.","url_abs":"https://arxiv.org/abs/2102.12293v3","url_pdf":"https://arxiv.org/pdf/2102.12293v3.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":"two-way-kernel-matrix-puncturing-towards","repo_url":"https://gricad-gitlab.univ-grenoble-alpes.fr/chatelaf/two-way-kernel-matrix-puncturing","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"}],"methods":[{"method_slug":"spectral-clustering","method_name":"Spectral Clustering"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2102.12293","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}