{"paper":{"title":"A Smooth Computational Transition in Tensor PCA","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.PR","stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Zhangsong Li","submitted_at":"2025-09-12T00:21:20Z","abstract_excerpt":"We propose an efficient algorithm for tensor PCA based on counting a specific family of weighted hypergraphs. For the order-$p$ tensor PCA problem where $p \\geq 3$ is a fixed integer, we show that when the signal-to-noise ratio is $\\lambda n^{-\\frac{p}{4}}$ where $\\lambda=\\Omega(1)$, our algorithm succeeds and runs in time $n^{C+o(1)}$ where $C=C(\\lambda)$ is a constant depending on $\\lambda$. This algorithm improves a poly-logarithmic factor compared to previous algorithms based on the Sum-of-Squares hierarchy \\cite{HSS15} or based on the Kikuchi hierarchy in statistical physics \\cite{WEM19}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.09904","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/2509.09904/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"}