{"paper":{"title":"Spectral Concentration and Recovery in Sparse High-Dimensional Random Geometric Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.PR","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Manuel Fernandez V, Yizhe Zhu","submitted_at":"2026-07-15T19:09:14Z","abstract_excerpt":"We study sparse random geometric graphs generated by connecting pairs of high-dimensional vectors whose inner product exceeds a threshold. The latent vectors are sampled either uniformly from the sphere or from a standard Gaussian distribution. Although every edge appears with probability $p$, the edges are dependent through their shared latent vectors. For the spherical model, at the connectivity scale $np=\\Omega(\\log n)$, we prove $\\|A-\\mathbb E A\\|=O\\left(\\sqrt{np\\log n}+np\\tau\\right)$, with high probability, where $\\tau$ is the cap threshold. This sharpens the spectral norm bound of Liu, M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14304","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.14304/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}