{"paper":{"title":"On Structural Decompositions of Finite Frames","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.FA","authors_text":"Alice Z.-Y. Chan, Allison Theobold, Logan Stokols, Martin S. Copenhaver, Sivaram K. Narayan","submitted_at":"2014-11-22T16:23:20Z","abstract_excerpt":"A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\\{f_i\\}_{i\\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\\{f_i\\}_{i\\in I}$ is said to be scalable if there exist nonnegative scalars $\\{c_i\\}_{i\\in I}$ such that $\\{c_if_i\\}_{i\\in I}$ is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame $\\{f_i\\}_{i\\in I}$ to be a collection of subsets of $I$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6138","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}