An improved bound on correlation between polynomials over Z_m and MOD_q.
Bibliographic record
Abstract
achatt3cs.mcgill.ca Let m, q> 1 be two integers that are co-prime and A be any subset of Zm. Let P be any multi-variate polynomial of degree d in n variables over Zm. We show that the MODq boolean function on n variables has correlation at most exp(−Ω(n/(m2 m−1)d)) with the boolean function f defined by f(x) = 1 iff P (x) ∈ A for all x ∈ {0, 1}n. This improves on the bound of exp(−Ω(n/(m2m)d)) obtained in the breakthrough work of Bourgain [3] and Green et al. [9]. Our calculation is also slightly shorter than theirs. Our result immediately implies the bound of exp(−Ω(n/4d)) for the special case of m = 2. This bound was first reported in the recent work of Viola [11]. [11] states that it is not clear how to extend their method to general m. 1
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.035 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.018 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".