{"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/learning-graphs-with-monotone-topology","title":"Learning Graphs with Monotone Topology Properties and Multiple Connected Components","arxiv_id":"1705.10934","date":"2017-05-31","proceeding":null,"authors":["Eduardo Pavez","Hilmi E. Egilmez","Antonio Ortega"],"abstract":"Recent papers have formulated the problem of learning graphs from data as an\ninverse covariance estimation with graph Laplacian constraints. While such\nproblems are convex, existing methods cannot guarantee that solutions will have\nspecific graph topology properties (e.g., being $k$-partite), which are\ndesirable for some applications. In fact, the problem of learning a graph with\ngiven topology properties, e.g., finding the $k$-partite graph that best\nmatches the data, is in general non-convex. In this paper, we develop novel\ntheoretical results that provide performance guarantees for an approach to\nsolve these problems. Our solution decomposes this problem into two\nsub-problems, for which efficient solutions are known. Specifically, a graph\ntopology inference (GTI) step is employed to select a feasible graph topology,\ni.e., one having the desired property. Then, a graph weight estimation (GWE)\nstep is performed by solving a generalized graph Laplacian estimation problem,\nwhere edges are constrained by the topology found in the GTI step. Our main\nresult is a bound on the error of the GWE step as a function of the error in\nthe GTI step. This error bound indicates that the GTI step should be solved\nusing an algorithm that approximates the similarity matrix by another matrix\nwhose entries have been thresholded to zero to have the desired type of graph\ntopology. The GTI stage can leverage existing methods (e.g., state of the art\napproaches for graph coloring) which are typically based on minimizing the\ntotal weight of removed edges. Since the GWE stage is formulated as an inverse\ncovariance estimation problem with linear constraints, it can be solved using\nexisting convex optimization methods. We demonstrate that our two step approach\ncan achieve good results for both synthetic and texture image data.","url_abs":"http://arxiv.org/abs/1705.10934v4","url_pdf":"http://arxiv.org/pdf/1705.10934v4.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":"learning-graphs-with-monotone-topology","repo_url":"https://github.com/STAC-USC/graph_learning_properties","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}