{"paper":{"title":"The Geometry of Nodal Sets and Outlier Detection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.FA","math.MP","stat.ML"],"primary_cat":"math.SP","authors_text":"Gal Mishne, Stefan Steinerberger, Xiuyuan Cheng","submitted_at":"2017-06-05T15:04:36Z","abstract_excerpt":"Let $(M,g)$ be a compact manifold and let $-\\Delta \\phi_k = \\lambda_k \\phi_k$ be the sequence of Laplacian eigenfunctions. We present a curious new phenomenon which, so far, we only managed to understand in a few highly specialized cases: the family of functions $f_N:M \\rightarrow \\mathbb{R}_{\\geq 0}$ $$ f_N(x) = \\sum_{k \\leq N}{ \\frac{1}{\\sqrt{\\lambda_k}} \\frac{|\\phi_k(x)|}{\\|\\phi_k\\|_{L^{\\infty}(M)}}}$$ seems strangely suited for the detection of anomalous points on the manifold. It may be heuristically interpreted as the sum over distances to the nearest nodal line and potentially hints at "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01362","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"}