{"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/locality-regularized-reconstruction","title":"Locality Regularized Reconstruction: Structured Sparsity and Delaunay Triangulations","arxiv_id":"2405.00837","date":"2024-05-01","proceeding":null,"authors":["Marshall Mueller","James M. Murphy","Abiy Tasissa"],"abstract":"Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points $[\\mathbf{x}_1, \\mathbf{x}_2, \\ldots, \\mathbf{x}_n] = \\mathbf{X} \\in \\mathbb{R}^{d \\times n}$ and a vector $\\mathbf{y} \\in \\mathbb{R}^d$, the goal is to find coefficients $\\mathbf{w} \\in \\mathbb{R}^n$ so that $\\mathbf{X} \\mathbf{w} \\approx \\mathbf{y}$, subject to some desired structure on $\\mathbf{w}$. In this work we seek $\\mathbf{w}$ that forms a local reconstruction of $\\mathbf{y}$ by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of $\\mathbf{X}$ that are close to $\\mathbf{y}$ when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of $\\mathbf{X}$ have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by $d+1$, thereby providing local sparse solutions when $d \\ll n$. Under the same condition we also show that for any $\\mathbf{y}$ contained in the convex hull of $\\mathbf{X}$ there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing $\\mathbf{y}$. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of $\\mathbf{X}$. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex.","url_abs":"https://arxiv.org/abs/2405.00837v1","url_pdf":"https://arxiv.org/pdf/2405.00837v1.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":"locality-regularized-reconstruction","repo_url":"https://github.com/MarshMue/LocalityRegularization","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"representation-learning","task_name":"Representation Learning"}],"methods":[{"method_slug":"set","method_name":"SET"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}