{"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/hilbert-exclusion-improved-metric-search","title":"Hilbert Exclusion: Improved Metric Search through Finite Isometric Embeddings","arxiv_id":"1604.08640","date":"2016-04-28","proceeding":null,"authors":["Connor Richard","Cardillo Franco Alberto","Vadicamo Lucia","Rabitti Fausto"],"abstract":"Most research into similarity search in metric spaces relies upon the\ntriangle inequality property. This property allows the space to be arranged\naccording to relative distances to avoid searching some subspaces. We show that\nmany common metric spaces, notably including those using Euclidean and\nJensen-Shannon distances, also have a stronger property, sometimes called the\nfour-point property: in essence, these spaces allow an isometric embedding of\nany four points in three-dimensional Euclidean space, as well as any three\npoints in two-dimensional Euclidean space. In fact, we show that any space\nwhich is isometrically embeddable in Hilbert space has the stronger property.\nThis property gives stronger geometric guarantees, and one in particular, which\nwe name the Hilbert Exclusion property, allows any indexing mechanism which\nuses hyperplane partitioning to perform better. One outcome of this observation\nis that a number of state-of-the-art indexing mechanisms over high dimensional\nspaces can be easily extended to give a significant increase in performance;\nfurthermore, the improvement given is greater in higher dimensions. This\ntherefore leads to a significant improvement in the cost of metric search in\nthese spaces.","url_abs":"http://arxiv.org/abs/1604.08640v1","url_pdf":"http://arxiv.org/pdf/1604.08640v1.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":"hilbert-exclusion-improved-metric-search","repo_url":"https://github.com/elki-project/elki","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}