Nondeterministic Interactive Refutations for Nearest Boolean Vector
Bibliographic record
Abstract
Most n-dimensional subspaces 𝒜 of ℝ^m are Ω(√m)-far from the Boolean cube {-1, 1}^m when n < cm for some constant c > 0. How hard is it to certify that the Nearest Boolean Vector (NBV) is at least γ √m far from a given random 𝒜? Certifying NBV instances is relevant to the computational complexity of approximating the Sherrington-Kirkpatrick Hamiltonian, i.e. maximizing x^TAx over the Boolean cube for a matrix A sampled from the Gaussian Orthogonal Ensemble. The connection was discovered by Mohanty, Raghavendra, and Xu (STOC 2020). Improving on their work, Ghosh, Jeronimo, Jones, Potechin, and Rajendran (FOCS 2020) showed that certification is not possible in the sum-of-squares framework when m ≪ n^1.5, even with distance γ = 0. We present a non-deterministic interactive certification algorithm for NBV when m ≫ n log n and γ ≪ 1/mn^1.5. The algorithm is obtained by adapting a public-key encryption scheme of Ajtai and Dwork.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".