{"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/uniform-embeddings-for-robinson-similarity","title":"Uniform Embeddings for Robinson Similarity Matrices","arxiv_id":"2105.09197","date":"2021-05-19","proceeding":null,"authors":["Jeannette Janssen","Zhiyuan Zhang"],"abstract":"A Robinson similarity matrix is a symmetric matrix where the entry values on all rows and columns increase toward the diagonal. Decompose the Robinson matrix into the sum of k {0, 1}-matrices, then these k {0, 1}-matrices are the adjacency matrices of a set of nested unit interval graphs. Previous studies show that unit interval graphs coincide with indifference graphs. An indifference graph has an embedding that maps each vertex to a real number, where two vertices are adjacent if their embedding is within a fixed threshold distance. In this thesis, consider k different threshold distances, we study the problem of finding an embedding that, simultaneously and with respect to each threshold distance, embeds the k indifference graphs corresponding to the k adjacency matrices. This is called a uniform embedding of a Robinson matrix with respect to the k threshold distances. We give a sufficient and necessary condition on Robinson matrices that have a uniform embedding, which is derived from paths in an associated graph. We also give an efficient combinatorial algorithm to find a uniform embedding or give proof that it does not exist, for the case where k = 2.","url_abs":"https://arxiv.org/abs/2105.09197v1","url_pdf":"https://arxiv.org/pdf/2105.09197v1.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"uniform-embeddings-for-robinson-similarity","repo_url":"https://github.com/OwenZZY/Robinson","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"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}