{"paper":{"title":"Metric recovery from directed unweighted graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SI","math.ST","stat.ME","stat.TH"],"primary_cat":"stat.ML","authors_text":"Tatsunori B. Hashimoto, Tommi S. Jaakkola, Yi Sun","submitted_at":"2014-11-20T23:16:09Z","abstract_excerpt":"We analyze directed, unweighted graphs obtained from $x_i\\in \\mathbb{R}^d$ by connecting vertex $i$ to $j$ iff $|x_i - x_j| < \\epsilon(x_i)$. Examples of such graphs include $k$-nearest neighbor graphs, where $\\epsilon(x_i)$ varies from point to point, and, arguably, many real world graphs such as co-purchasing graphs. We ask whether we can recover the underlying Euclidean metric $\\epsilon(x_i)$ and the associated density $p(x_i)$ given only the directed graph and $d$.\n  We show that consistent recovery is possible up to isometric scaling when the vertex degree is at least $\\omega(n^{2/(2+d)}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5720","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"}