{"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/laplacian-eigenmaps-and-spectral-techniques","title":"Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering","arxiv_id":null,"date":"2020-05-13","proceeding":"‏‏‎ ‎ 2020 5","authors":["Mikhail Belkin and Partha Niyogi"],"abstract":"One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.","url_abs":"https://ieeexplore.ieee.org/abstract/document/6789755","url_pdf":"https://www2.imm.dtu.dk/projects/manifold/Papers/Laplacian.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":"laplacian-eigenmaps-and-spectral-techniques","repo_url":"https://github.com/benedekrozemberczki/karateclub","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"dimensionality-reduction","task_name":"Dimensionality Reduction"}],"methods":[{"method_slug":"lapeigen","method_name":"LapEigen"}],"datasets_introduced":[],"methods_introduced":[{"slug":"lapeigen","name":"LapEigen","full_name":"Laplacian EigenMap"}],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}