{"paper":{"title":"The Generalized Schur Decomposition and the rank-$R$ set of real $I\\times J\\times 2$ arrays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alwin Stegeman","submitted_at":"2010-11-15T15:58:19Z","abstract_excerpt":"It is known that a best low-rank approximation to multi-way arrays or higher-order tensors may not exist. This is due to the fact that the set of multi-way arrays with rank at most $R$ is not closed. Nonexistence of the best low-rank approximation results in diverging rank-1 components when an attempt is made to compute the approximation. Recently, a solution to this problem has been proposed for real $I\\times J\\times 2$ arrays. Instead of a best rank-$R$ approximation the best fitting Generalized Schur Decomposition (GSD) is computed. Under the restriction of nonsingular upper triangular matr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3432","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"}