{"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/the-neighborhood-lattice-for-encoding-partial","title":"The neighborhood lattice for encoding partial correlations in a Hilbert space","arxiv_id":"1711.00991","date":"2017-11-03","proceeding":null,"authors":["Arash A. Amini","Bryon Aragam","Qing Zhou"],"abstract":"Neighborhood regression has been a successful approach in graphical and\nstructural equation modeling, with applications to learning undirected and\ndirected graphical models. We extend these ideas by defining and studying an\nalgebraic structure called the neighborhood lattice based on a generalized\nnotion of neighborhood regression. We show that this algebraic structure has\nthe potential to provide an economic encoding of all conditional independence\nstatements in a Gaussian distribution (or conditional uncorrelatedness in\ngeneral), even in the cases where no graphical model exists that could\n\"perfectly\" encode all such statements. We study the computational complexity\nof computing these structures and show that under a sparsity assumption, they\ncan be computed in polynomial time, even in the absence of the assumption of\nperfectness to a graph. On the other hand, assuming perfectness, we show how\nthese neighborhood lattices may be \"graphically\" computed using the separation\nproperties of the so-called partial correlation graph. We also draw connections\nwith directed acyclic graphical models and Bayesian networks. We derive these\nresults using an abstract generalization of partial uncorrelatedness, called\npartial orthogonality, which allows us to use algebraic properties of\nprojection operators on Hilbert spaces to significantly simplify and extend\nexisting ideas and arguments. Consequently, our results apply to a wide range\nof random objects and data structures, such as random vectors, data matrices,\nand functions.","url_abs":"http://arxiv.org/abs/1711.00991v2","url_pdf":"http://arxiv.org/pdf/1711.00991v2.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":"the-neighborhood-lattice-for-encoding-partial","repo_url":"https://github.com/aaamini/reg_lattice_comp","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"dimensionality-reduction","task_name":"Dimensionality Reduction"},{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"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}