{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"dd1f1eb3c8350dd38bb4f84b076291fb56a5ccabcd66c04cc92588fc00eeb232","created":"2026-10-03T07:18:11Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"c6a525b2347fa57b1078404477c112713b00fe21abc0539e065edb4f9d08b40b","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"stat.learning-inference.low-degree-conjecture-repair","kind":"well-posed","literature_status":"contested","n":"1","parents":[],"plain":"A popular test predicts that a hidden signal cannot be found quickly whenever no low-degree polynomial (a simple formula built from products of a few data entries) can detect it. Recent counterexamples show this test fails in some cases, and the task is to find the exact conditions under which it is right.","posed_since":"2018","precise":"For a planted distribution $P_n$ and a null distribution $Q_n$ on $\\{0,1\\}^M$ (for example random graphs), the degree-D advantage is the best correlation of a degree-D polynomial with the likelihood ratio $\\mathrm{d}P_n/\\mathrm{d}Q_n$. Identify an explicit, checkable condition on $(P_n, Q_n)$ such that a bounded advantage at $D \\sim \\operatorname{polylog}(n)$, together with the condition, implies that no polynomial-time algorithm distinguishes $P_n$ from $Q_n$ after independent noise; or prove that no condition covering standard planted problems (planted clique, sparse PCA, tensor PCA, community detection) can exist. An answer is a stated corrected conjecture with a proof, or a proof that the program fails.","problem_ref":null,"references":"","settled_by":"A theorem establishing a corrected low-degree hardness criterion for a class containing the standard planted problems, or a counterexample within that class.","status_note":"Holmgren and Wein (2020) gave counterexamples lacking symmetry, which motivated the permutation-invariance and noise conditions; Buhai, Hsieh, Jain and Kothari (2025, arXiv:2505.17360) refuted the quasi-polynomial version of Hopkins' conjecture, and Mao (July 2026 preprint, arXiv:2607.20318, not yet refereed) reports a counterexample to the polynomial-time version for permutation-invariant graph distributions; both are permutation-invariant and stable under noise, so a valid criterion needs a further condition.","title":"A correct low-degree criterion for polynomial-time hardness of 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